2.1. Constructing WRTs from URTs

We will first introduce a coupling allowing us to construct a special kind of WRT from a URT. We will not use this coupling in the analysis of random tree statistics because the second coupling construction introduced below can be applied to a more general class of WRTs. We still wanted to discuss this version because it has URTs rather than Hoppe trees as a starting point, and URTs are a much better studied structure.

We will call trees that have a weight sequence such that the first k nodes have a constant weight equal to $\theta$ and all other nodes have weight 1, $\theta^k$ -RTs. It is possible to construct a $\theta^k$ -RT from a URT by a coupling construction when $\theta \in \mathbb{N}^+$ . To emphasize this assumption we will use m instead of $\theta$ in this part. In particular, we can construct Hoppe trees for which the weight of the root is a natural number by this coupling. To avoid confusion let us denote the nodes in the URT by i and the nodes in the $m^k$ -RT obtained by the coupling by $i^*$ .

The coupling construction in that case goes as follows: First construct a URT with $mk + n-k$ nodes. We write $\mathcal{T}_{mk+n-k}$ for this URT and $\mathcal{T}_n^{m^k}$ for the tree that we obtain at the end of the coupling. Since we want the weight of the first k nodes to be m, we need to join several nodes into one. More precisely, we combine m nodes of the URT to get each of the first k nodes of the $m^k$ -RT in the following way:

1. • Nodes $1, \dots, m$ become node $1^*$ ;

2. • $m+1, \dots, 2m$ become node $2^*$ ;

3. • $(k-1)m +1, \dots, km$ become node $k^*$ .

All nodes after km in $\mathcal{T}_{mk+n-k}$ correspond only to a single node in $\mathcal{T}_n^{m^k}$ ; we just need to ‘translate’ the names of the nodes to take into account that we used mk nodes instead of k for the first k nodes in the reconstructed tree. More precisely, for $j>km$ , node j in the URT becomes node $(\,j-k(m-1))^*$ in the WRT.

The parent–child relations in the WRT follow from those in the URT as follows: If, for $j>km$ and $i \leq k $ , j is a child of one of the nodes $(i-1)m+1, \dots, im$ in the URT, $(\,j-k(m-1))^*$ becomes a child of $i^*$ in the WRT. If, for $j>km$ , the parent of node j is h with $h>km$ , the parent of node $(\,j-k(m-1))^*$ is node $(h-k(m-1))^*$ .

For the nodes before or equal to km, we need to be more careful, since we joined several nodes into one and the nodes $(\,j-1)m +1, \dots, jm$ might have different parents in the URT. To settle this ambiguity, for $1<j\leq k$ we set the parent of $j^*$ as the parent of $(\,j-1)m+1$ , i.e. of the node with the smallest label among those that become $j^*$ . Thus, if in the URT the parent of $(\,j-1)m+1$ is any of the nodes $(i-1)m+1, \dots, im$ , the parent of $j^*$ is $i^*$ . It can be easily verified that the attachment probabilities obtained correspond to those of an $m^k$ -tree; for details, see [Reference Hiesmayr9].

In Figure 1 we present a small example of the coupling construction. In the example, nodes 1 and 2 combine to form node $1^*$ , and nodes 3 and 4 become node $2^*$ . Then nodes 5 and 6 become node $3^*$ , and $3^*$ is a child of $2^*$ , since node 5 is a child of node 4 in the URT. Nodes 7 and 8 become node $4^*$ , and $4^*$ is a child of $1^*$ since node 7 is a child of node 1 in the URT. Finally, node 9 becomes node $5^*$ , and $5^*$ is a child of $4^*$ since node 9 is a child of node 7 in the URT.

Figure 1:

A WRT $\mathcal{T}_5^{2^4}$ is constructed from a URT $\mathcal{T}_9$ .

We now show, as an example, how the coupling can be used to study the number of leaves, i.e. the number of nodes without children, of an $m^k$ -RT. Let $\mathcal{L}_{mk+n-k}$ denote the number of leaves of $\mathcal{T}_{mk+n-k}$ , and $\mathcal{L}_n^{m^k}$ denote the number of leaves of $\mathcal{T}_{n}^{m^k}$ . Then $\mathcal{L}_{mk+n-k}$ can be used to bound $\mathcal{L}_n^{m^k}$ . First, if node $i > km $ is a leaf in $\mathcal{T}_{mk+n-k}$ , the corresponding node in $\mathcal{T}_n^{m^k}$ , which is ${(i-k(m-1))}^*$ , is also a leaf. The reconstruction process thus only affects the children of the nodes $i^*$ with $1 \leq i \leq k$ , and the root cannot be a leaf, so we can have at most $k-1$ additional leaves. Moreover, for each $1 \leq i \leq k$ we can ‘lose’ at most $m-1$ leaves since if all of $(i\,{-}\,1)m+1, \dots, im$ are leaves in $\mathcal{T}_{mk+n-k}$ , $i^*$ will be a leaf in $\mathcal{T}_n^{m^k}$ too. Hence, we can conclude that

\begin{equation*}\mathcal{L}_{mk+n-k}-k(m-1)< \mathcal{L}_n^{m^k} <\mathcal{L}_{mk+n-k}+ k-1 .\end{equation*}

Together with results about the expected number of leaves of URTs, this implies, after some simple manipulations, that

\begin{equation*}\begin{split}\left |\mathrm{E}[\mathcal{L}_n^{m^k}]-\mathrm{E}[\mathcal{L}_n] \right | \leq \frac{k(m+1)}{2}, \\\end{split}\end{equation*}

where $\mathcal{L}_n$ denotes the number of leaves in a URT on n nodes. It is possible to derive other results from this coupling, but since the coupling we present below is more general, we will not go further into it here.

2.2. Constructing WRTs from Hoppe trees

We will now introduce a coupling construction for WRTs whose nodes have constant weight after a certain index, a class similar to, but more general, than $\theta^k$ -RTs. Let $\mathcal{T}_n^w$ be a WRT, and $(w_i)_{i \in \mathbb{N}}$ , the weight sequence of $\mathcal{T}_n^w$ , be such that there is a $k \in \mathbb{N}$ such that $w_i=1$ for all $i>k$ . Then we can construct $\mathcal{T}_n^w$ from a Hoppe tree with root weight $\theta=\sum_{i=1}^{k} w_i$ by a coupling construction, more precisely by splitting the root into k nodes. To avoid confusion we will write i for node i in the Hoppe tree and $i^*$ for node i in the reconstructed tree.

We now describe the coupling construction: First construct a Hoppe tree on $n-k+1$ nodes and with $\theta$ , the weight of the root, equal to $\sum_{i=1}^{k} w_i$ . Then construct a WRT of size k corresponding to $(w_i)_{i \in \{1,\dots,k\}}$ . Now we replace the root of the Hoppe tree by this weighted recursive tree of size k in the following way: nodes $1^*$ , $2^*$ , …, $k^*$ are the nodes of the WRT of size k we just constructed. For $i \geq 2$ , node i in the Hoppe tree becomes node $(i+k-1)^*$ in the reconstructed tree, so we shift the names of the rest of the nodes by $k-1$ . Then, for all $i \geq 2$ , if i is a child of node 1 in the Hoppe tree, $(i+k-1)^*$ becomes a child of one of the nodes $1^*, \dots, k^*$ in the reconstructed tree, proportionally to their weights. This means that if i is a child of node 1 in the Hoppe tree, for $1\leq j^* \leq k$ , node $(i+k-1)^*$ will become a child of a node $j^*$ in the reconstructed tree with probability $\frac{w_{j}}{\sum_{\ell=1}^{k} w_{\ell}}$ .

Let us check that this gives the attachment probabilities corresponding to the WRT we aim to construct.

For $1\leq i^* \leq k<j^*$ ,

\begin{equation*}\begin{split}\mathrm{P} & (\,j^* \text{ is child of } i^*) = \mathrm{P}(\,j-k+1 \text{ is child of } 1, \, i^*\text{ is chosen as the parent of } j^*) \\&= \frac{\sum_{\ell=1}^{k} w_{\ell}}{j-k+1 -2 +\sum_{\ell=1}^{k} w_{\ell}} \frac{w_i}{\sum_{\ell=1}^{k} w_{\ell}} = \frac{w_i}{j-1 -k+\sum_{\ell=1}^{k} w_{\ell} }. \\\end{split}\end{equation*}

For $k<i^*<j^*$ ,

\begin{equation*}\begin{split}\mathrm{P} & (\,j^* \text{ is child of } i^*)= \mathrm{P}(\,j-k+1 \text{ is child of } i-k+1)\\&= \frac{1}{j-k+1 -2 +\sum_{\ell=1}^{k} w_{\ell}} = \frac{1}{j-1-k+\sum_{\ell=1}^{k} w_{\ell}}. \\\end{split}\end{equation*}

The following two sections will be using the coupling construction just described.

3.1. Number of leaves

A node in a tree with degree one is said to be a leaf. Focusing on the number of leaves, the reconstruction process in our coupling construction implies that all the leaves of the Hoppe tree are still leaves in the reconstructed tree, since we do not change any relation among the nodes $2, \dots, n-k+1$ of the Hoppe tree or, respectively, nodes $(k+1)^*, \dots, n^*$ of the reconstructed tree. There can be at most $k-1$ additional leaves among the first k nodes. Thus, we can bound the number of leaves $ \mathcal{L}_n^{w} $ of $\mathcal{T}^w$ by the number of leaves $\mathcal{L}_{n}^{\theta}$ of $\mathcal{T}^\theta$ :

(1) \begin{equation} {\mathcal{L}_{n-k+1}^{\theta} \leq \mathcal{L}_n^{w} \leq \mathcal{L}_{n-k+1}^{\theta} +k-1. }\end{equation}

In [Reference Leckey and Neininger10] the following results about the leaves of Hoppe trees are given. In this theorem and below, $\mathcal{G}$ denotes a standard normal random variable.

Theorem 1. ([Reference Leckey and Neininger10].) Let $\mathcal{L}_n^{\theta}$ denote the number of leaves of a Hoppe tree with $n \geq 2 $ nodes. Then

\begin{equation*}\begin{split}&\mathrm{E} [\mathcal{L}_n^{\theta}] = \frac{n}{2} + \frac{\theta -1}{2} + \mathcal{O}\bigg(\frac{1}{n} \bigg), \\[2pt]&\mathrm{Var} (\mathcal{L}_n^{\theta}) = \frac{n}{12} + \frac{\theta -1}{12} + \mathcal{O}\bigg(\frac{1}{n} \bigg), \\[2pt]& \mathrm{P}(|\mathcal{L}_n^{\theta} - \mathrm{E}[\mathcal{L}_n^{\theta}] | \geq t) \leq 2 \exp\bigg\{-\frac{6t^2}{n+\theta+1}\bigg\} \quad \text{ for all } t>0, \text{ and } \\[2pt]&\frac{\mathcal{L}_n^{\theta} - \mathrm{E}[\mathcal{L}_n^{\theta}]}{\sqrt{\mathrm{Var}\left (\mathcal{L}_n^{\theta}\right)}} \longrightarrow_{\rm d} \mathcal{G} \text{ as } n \to \infty.\end{split}\end{equation*}

Using the above theorem we can derive results on the number of leaves in WRTs whose nodes have constant weight after a certain index.

Theorem 2. Let $\mathcal{L}^{w}_{n}$ denote the number of leaves of a WRT of size n with weight sequence $(w_i)_{i \in \mathbb{N}}$ such that there is a $k \in \mathbb{N}$ such that for all $i>k$ we have $w_i = 1$ . Then

\begin{equation*}\begin{split}&\mathrm{E} [\mathcal{L}_n^{w}] = \frac{n}{2} + C + \mathcal{O} \left ( \frac{1}{n} \right ) \text{ with } |C| \leq \frac{\sum_{i=1}^{k} w_i +k}{2},\\[2pt] &\mathrm{Var}(\mathcal{L}_n^{w}) = \frac{n}{12} + \mathcal{O} (\sqrt{n}), \\[2pt] & \mathrm{P} (|\mathcal{L}_n^{w} - \mathrm{E}[\mathcal{L}_n^{w}] | \geq t) \leq 2 \exp\Bigg\{- \frac{6(t-2k+2)^2}{n-k+2+ \sum_{i=1}^k w_i}\Bigg\} \quad \text{ for all } t>2k-2, \text{ and } \\[2pt] & \frac{\mathcal{L}^{w}_{n} - \mathrm{E}[\mathcal{L}^{w}_{n}]}{\sqrt{\mathrm{Var}\left (\mathcal{L}_n^{w}\right)}} \longrightarrow_{\rm d} \mathcal{G} \text{ as } n \to \infty.\end{split}\end{equation*}

Proof. First, for the expected value we get, from Theorem 1 and (1),

\begin{align*}&\mathcal{L}_{n-k+1}^{\theta} \leq \mathcal{L}_n^{w} \leq \mathcal{L}_{n-k+1}^{\theta} +k-1 \\[2pt]&\quad \Rightarrow \frac{n-k+1}{2} + \frac{\sum_{i=1}^{k} w_i-1}{2} + \mathcal{O}\left ( \frac{1}{n} \right ) \leq \mathrm{E}\left [\mathcal{L}_n^{w}\right ] \\[2pt]&\qquad \leq \frac{n-k+1}{2} + \frac{\sum_{i=1}^{k} w_i-1}{2} + k -1 + \mathcal{O}\left ( \frac{1}{n} \right ) \\[2pt]&\quad \Rightarrow \frac{n}{2} + \frac{\sum_{i=1}^{k} w_i-k}{2} + \mathcal{O}\left ( \frac{1}{n} \right ) \leq \mathrm{E} \left [\mathcal{L}_n^{w}\right ] \leq \frac{n}{2} + \frac{\sum_{i=1}^{k} w_i+k-2}{2} + \mathcal{O}\left ( \frac{1}{n} \right ) \\[2pt]&\quad \Rightarrow \mathrm{E}\left [\mathcal{L}_n^{w}\right ] = \frac{n}{2} + C + \mathcal{O}\left ( \frac{1}{n} \right ),\end{align*}

where C depends on k and $w_1, \dots, w_k$ , and we have $|C| \leq \frac{\sum_{i=1}^{k} w_i +k}{2}$ .

Equation (1) also directly gives a concentration inequality. We have that

\begin{equation*}\begin{split}\mathrm{P} & (|\mathcal{L}_n^{w} - \mathrm{E}[\mathcal{L}_n^{w}]|\geq t) \\[2pt] & \leq \mathrm{P}( |\mathcal{L}_n^{w} - \mathcal{L}_{n-k+1}^{\theta} | + |\mathcal{L}_{n-k+1}^{\theta} - \mathrm{E} [\mathcal{L}_{n-k+1}^{\theta} ] | + | \mathrm{E} [\mathcal{L}_{n-k+1}^{\theta} ] - \mathrm{E} [\mathcal{L}_n^{w} ] |\geq t) \\[2pt] & \leq \mathrm{P}(|\mathcal{L}_{n-k+1}^{\theta} - \mathrm{E} [\mathcal{L}_{n-k+1}^{\theta} ] | \geq t-2k+2) \\[2pt] & \leq 2 \exp\Bigg\{- \frac{6(t-2k+2)^2}{n-k+2+ \sum_{i=1}^k w_i}\Bigg\}.\end{split}\end{equation*}

The coupling also gives us an approximation for the variance. Let Y denote the number of additional leaves among the first k nodes in the reconstructed tree. Then $\mathrm{Var}(Y) = { \mathcal{O}(k^2)}$ since $Y \leq k$ . Since $\mathrm{Var}(\mathcal{L}_n^{w}) = \mathrm{Var}(\mathcal{L}_{n-k+1}^{\theta} +Y ) = \mathrm{Var}(\mathcal{L}_{n-k+1}^{\theta}) + \mathrm{Var}(Y) + { 2 \mathrm{Cov}(\mathcal{L}_{n-k+1}^{\theta} ,Y)}$ , we get, by the Cauchy–Schwarz inequality,

\begin{equation*}\mathrm{Var}(\mathcal{L}_n^{w}) = \frac{n}{12} + \mathcal{O} (\sqrt{n}).\end{equation*}

In a similar way, one can make conclusions about the asymptotic distribution. For this we will need Slutsky’s theorem: Let $X_n$ and $Y_n$ be sequences of random variables such that $X_n \to_{\rm d} X$ and $Y_n \to c$ in probability for $c \in \mathbb{R}$ . Then

\begin{equation*} \lim_{n\to \infty} X_n + Y_n =_{\rm d} X +c.\end{equation*}

In order to derive a central limit theorem for $\mathcal{L}^{w}_{n}$ , we write

\begin{align*}\frac{\mathcal{L}^{w}_{n}- \mathrm{E}[\mathcal{L}_{n}^w]}{\sqrt{\mathrm{Var}(\mathcal{L}_{n}^w)}}&= \frac{\mathcal{L}^{w}_{n}- \mathcal{L}_{n-k+1}^\theta}{\sqrt{\mathrm{Var}(\mathcal{L}_{n}^w)}}+ \frac{\mathcal{L}^{\theta}_{n-k+1}- \mathrm{E}[\mathcal{L}_{n-k+1}^\theta]}{\sqrt{\mathrm{Var}(\mathcal{L}_{n}^w)}} \\[2pt]&\quad + \frac{\mathrm{E}[\mathcal{L}^{\theta}_{n-k+1}]- \mathrm{E}[\mathcal{L}_{n}^w]}{\sqrt{\mathrm{Var}(\mathcal{L}_{n}^w)}}.\end{align*}

Now, by Theorem 1 and our previous result on $\mathrm{Var}(\mathcal{L}_n^w)$ we have

\begin{equation*}\frac{\mathcal{L}^{\theta}_{n-k+1}- \mathrm{E}[\mathcal{L}_{n-k+1}^\theta]}{\sqrt{\mathrm{Var}(\mathcal{L}_{n-k+1}^\theta)}} \sqrt{\frac{\mathrm{Var}(\mathcal{L}_{n-k+1}^\theta)}{\mathrm{Var}(\mathcal{L}_{n}^w)}} \longrightarrow_{\rm d}\mathcal{G} \quad \text{as } {n \to \infty} ,\end{equation*}

and since $\mathrm{Var}(\mathcal{L}_{n}^w) \longrightarrow \infty$ as $n \to \infty$ we have, by (1), that

\begin{equation*}\left | \frac{\mathcal{L}^{w}_{n}- \mathcal{L}_{n-k+1}^\theta}{\sqrt{\mathrm{Var}(\mathcal{L}_{n}^w)}} + \frac{\mathrm{E}[\mathcal{L}^{\theta}_{n-k+1}]- \mathrm{E}[\mathcal{L}_{n}^w]}{\sqrt{\mathrm{Var}(\mathcal{L}_{n}^w)}} \right | \leq \frac{2k}{\sqrt{\mathrm{Var}(\mathcal{L}_{n}^w)}} \rightarrow 0 \text{ almost surely}.\end{equation*}

Now we can apply Slutsky’s theorem and conclude that

\begin{align*} \frac{\mathcal{L}^{w}_{n}- \mathrm{E}[\mathcal{L}_{n}^w]}{\sqrt{\mathrm{Var}(\mathcal{L}_{n}^w)}} \rightarrow_{\rm d} \mathcal{G} \quad \text{as } {n \to \infty}.\\[-30pt] \end{align*}

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Remark 1. It might be possible to get results on the number of leaves of a general WRT $\mathcal{T}_n^w$ by writing $\mathcal{L}_{n}^{w}$ as the sum of $\textbf{1}(\ell_i^{w})$ , where $\ell_i^{w}$ denotes the event that node i is a leaf in $\mathcal{T}_n^w$ . It follows from the construction principle that

\begin{equation*}\mathrm{P} \left (\ell_i^{w}\right ) = \prod_{j=i+1}^{n} \left ( 1- \frac{w_i}{w_1 + \dots + w_{j-1}} \right ).\end{equation*}

After some manipulation this expression becomes

\begin{equation*}\mathrm{P} \left (\ell_i^{w}\right ) = \frac{ w_1 + \dots + w_{i-1}}{w_1 + \dots + w_{n-1}} \prod_{j=i+1}^{n-1} \left ( 1 + \frac{w_{j} - w_{i}}{w_1 + \dots + w_{j-1}} \right ).\end{equation*}

An exact expression for the expectation of the number of leaves of a $\theta^k$ -RT can be obtained by writing

\begin{equation*}\mathrm{E}\left[\mathcal{L}_n^{\theta^k}\right ] = \sum_{i=2}^{n} \mathrm{E}\left [\textbf{1}\left (\ell_i^{\theta^k}\right )\right ]\end{equation*}

and using the expression above. After some computations we get

\begin{equation*}\mathrm{E}\left [\mathcal{L}_n^{\theta^k} \right ] = \frac{n}{2} + \frac{k(\theta-1)}{2} + \frac{k \theta(1-k\theta)}{2(k (\theta-1) + n-1)}+ \frac{k-1}{2} \prod_{i=1}^{n-1-k} \frac{\theta (k-1)+i}{\theta k +i}.\end{equation*}

3.2. Height

As a second example we discuss the height of a WRT, which is defined as the length of the longest path from the root to a leaf. Let $\mathcal{H}_n^w$ denote the height of a WRT with weight sequence $(w_i)_{i \in \mathbb{N}}$ such that $w_i =1$ for $i>k$ . Let, moreover, $\mathcal{D}_{1,i}^w$ and $\mathcal{D}_{1,i}^\theta$ denote the distance between the root and node i in the reconstructed weighted tree and the original Hoppe tree respectively. For $i \leq k$ , $\mathcal{D}_{1,i}^w$ is at most $k-1$ . Also, for any $i>k$ , the path from the root to $i^*$ corresponds to the path from the root to the corresponding node in the original Hoppe tree, i.e. the distance from the root to $i-k+1$ , except that we might have an additional path among the first k nodes instead of the first edge. Thus, $\mathcal{D}^w_{1,i}$ is at least as big as $\mathcal{D}_{i-k+1}^\theta$ .

Also, $\mathcal{D}^w_{1,i}$ is at most $k-1$ bigger than the distance between the root and the corresponding node in the original tree. Let $j-k+1$ be the first node on the path from node 1 to $i-k+1$ in the original tree. Then, in the reconstructed tree $j^*$ will be attached to some $h^*$ , where $1\leq h \leq k$ . Thus, when we consider the path consisting of the path from the root to $h^*$ in the reconstructed tree, the edge from $h^*$ to $j^*$ and the path from $j-k+1$ to $i-k+1$ in the Hoppe tree, we get a path from $1^*$ to $i^*$ in the reconstructed tree. Thus, for all $k+1 \leq i \leq n$ , there is some $h \leq k$ such that

\begin{equation*}{D}^w_{1,i} = \mathcal{D}_{1,h}^w + 1 + \mathcal{D}_{j-k+1,i-k+1}^\theta = \mathcal{D}_{1,h}^w + \mathcal{D}_{1,i-k+1}^\theta.\end{equation*}

Also, $\mathcal{D}_{1,h}^w \leq k-1$ , so we have

\begin{equation*}\mathcal{D}_{1,i-k+1}^\theta \leq {D}^w_{1,i} \leq \mathcal{D}_{1,i-k+1}^\theta +k-1,\end{equation*}

which implies that

\begin{equation*}\begin{split}\max_{i =1, \dots, n-k+1} \{\mathcal{D}_{1,i}^\theta\} \leq \max_{i =1, \dots, n} \{\mathcal{D}_{1,i}^w\} \leq \max_{i =1, \dots, n-k+1} \{\mathcal{D}_{1,i}^\theta\} + k-1.\end{split}\end{equation*}

Thus, from the definition of the height as $\mathcal{H}_n = \max_{i =1, \dots, n} \{\mathcal{D}_{1,i}\},$ we have

(2) \begin{equation} \begin{split}\mathcal{H}_{n-k+1}^\theta \leq \mathcal{H}_n^{w} \leq \mathcal{H}_{n-k+1}^\theta + k-1.\end{split}\end{equation}

We have the following result about the height of Hoppe trees.

Theorem 3. ([Reference Leckey and Neininger10].) Let $\mathcal{H}_n^{\theta}$ denote the height of a Hoppe tree with n nodes. Then

\begin{align*}\mathrm{E} [\mathcal{H}_n^{\theta}] &= {\rm e}\ln(n) - \frac{3}{2}\ln \ln n + \mathcal{O}(1) , \\[3pt] \mathrm{Var}(\mathcal{H}_n^{\theta}) &= \mathcal{O}(1).\end{align*}

Together with the coupling, this allows us to derive the following theorem.

Theorem 4. Let $\mathcal{H}^{w}_{n}$ denote the height of a WRT of size n with weight sequence $(w_i)_{i \in \mathbb{N}}$ such that there is a $k \in \mathbb{N}$ such that, for all $i>k$ , we have $w_i = 1$ . Then

\begin{align*}\mathrm{E}\left [\mathcal{H}_n^{w} \right ] &= {\rm e} \ln(n) - \frac{3}{2} \ln \ln (n) + \mathcal{O}(1) , \\[3pt] \mathrm{Var}(\mathcal{H}_n^{w}) &= \mathcal{O}(1).\end{align*}

Proof. According to Theorem 3 we have $\mathrm{E}[\mathcal{H}_{n-k+1}^{\theta}] = {\rm e}\ln(n-k+1) - \frac{3}{2}\ln \ln (n-k\,{+}\,1) + \mathcal{O}(1)$ . Note that

\begin{align*} {\rm e}\ln(n-k+1) - \frac{3}{2}\ln \ln (n-k+1) & = {\rm e}\Big( \ln(n) + \ln\Big(1-\frac{k-1}{n} \Big) \Big) \\& \quad - \frac{3}{2} \bigg( \ln \ln(n) + \ln \Big( 1+ \frac{\ln\big(1-\frac{k-1}{n} \big)}{\ln(n)} \Big) \bigg) \\& = {\rm e} \ln(n) - \frac{3}{2} \ln \ln (n) + o(1).\end{align*}

Thus, we get, by (2) for $n>k$ ,

\begin{equation*}{\rm e} \ln(n) - \frac{3}{2} \ln \ln (n) + \mathcal{O}(1) \leq \mathrm{E}\left [\mathcal{H}_n^{w} \right ] \leq {\rm e} \ln(n) - \frac{3}{2} \ln \ln (n) + \mathcal{O}(1)+ k-1 ,\end{equation*}

which implies

\begin{equation*}\mathrm{E}\left [\mathcal{H}_n^{w} \right ] = {\rm e} \ln(n) - \frac{3}{2} \ln \ln (n) + \mathcal{O}(1).\end{equation*}

As before, we might define $Y\coloneqq \mathcal{H}_n^{w} - \mathcal{H}_{n-k+1}^\theta$ . Then, by (2), we have $Y<k$ , so

\begin{align*}\mathrm{Var}(\mathcal{H}_n^{w}) &= \mathrm{Var}(\mathcal{H}_{n-k+1}^\theta+Y) \\[3pt] &= \mathrm{Var}(\mathcal{H}_{n-k+1}^\theta) + \mathrm{Var}(Y) + { 2 \mathrm{Cov}(\mathcal{H}_{n-k+1}^\theta, Y) } \\[3pt] &= \mathcal{O}(1).\\[-30pt] \end{align*}

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4.1. Permutations view

In this section we focus on standard Hoppe trees and study the size of the largest branch in this tree model. The results will sharpen the corresponding observations of [Reference Feng, Su and Hu6] for URTs. Before moving on to largest branches, we need to discuss constructions of Hoppe trees via random permutations, in particular permutations that are generated via the Ewens sampling formula.

For each n, the Ewens distribution gives a family of distributions over the vectors

\begin{equation*}C^{(n)} = (C_1^{(n)}, C_2^{(n)},\ldots,C_n^{(n)})\end{equation*}

with $\sum_{i=1}^n i C_i^{(n)} = n$ . In particular, for given $\theta > 0$ , the Ewens distribution ${\rm EW}(\theta)$ is defined with the probabilities

\begin{equation*}\mathrm{P}_{\theta} (C^{(n)} = (c_1,\ldots,c_n))= \textbf{1}\left(\sum_{j=1}^n j c_j = n \right) \frac{n!}{\theta_{(n)}} \prod_{j=1}^n \left( \frac{\theta}{j} \right)^{c_j} \frac{1}{c_j!}, \qquad c_1,\ldots,c_n \in \mathbb{N},\end{equation*}

where $\theta_{(n)} = \theta(\theta + 1) \cdots (\theta + n - 1)$ .

Below we call a permutation whose cycle structure results from ${\rm EW}(\theta)$ a Hoppe permutation. In this setting, $c_i$ stands for the number of cycles in the permutation of size i. In order to relate the topic to Hoppe trees, we need to discuss a recursive construction of Hoppe permutations. The discussion here is similar to the one in [Reference Altok and Işlak1]. For convenience, the permutation will be constructed on the label set $\{2,3,\ldots,n\}$ . We first begin with the permutation (2) with only one cycle. Then 3 either joins the first cycle to the right of 2 with probability $\frac{1}{\theta + 1}$ , or starts the second cycle with probability $\frac{\theta}{\theta + 1}$ . Once we have constructed a permutation on $\{2,3,\ldots,k-1\}$ , k either starts a new cycle with probability $\frac{\theta}{\theta +k - 2}$ , or is inserted to the right of a randomly chosen integer already assigned to a cycle. The resulting permutation then has the cycle structure from the distribution ${\rm EW}(\theta)$ [Reference Arratia, Barbour and Tavaré2].

Next, we construct a Hoppe tree based on the permutation construction of the previous paragraph. Begin with node 1 as the root and node 2 attached to it. Then, if node 3 begins a new cycle in the corresponding Hoppe permutation, attach it to node 1, and otherwise attach it to node 2. Then, for node $k \geq 4$ , if k starts a new cycle in the Hoppe permutation, attach node k to node 1, and otherwise attach it to node j where k was inserted to the right of j in the corresponding permutation.

It is then clear that this gives a bijection between Hoppe permutations and Hoppe trees. In particular, the cycles in a Hoppe permutation are in a one-to-one relation with the branches in the corresponding tree. This reduces the study of the size of the largest branch in a Hoppe tree to the study of the largest cycle in its permutation correspondence.

4.2. Size of the largest branch

For a given tree $\mathcal{T}$ on n vertices, the number of branches is the number of children of the root, i.e. all nodes that are attached to the root. If a node i is attached to the root, then node i with its descendants is said to form a branch of the tree. Let $\mathcal{B}_{n,i}(\mathcal{T})$ be the number of branches of size i in $\mathcal{T}$ . Also define

\begin{equation*}\nu_n(\mathcal{T}) \coloneqq \max\{i \in [n-1]\,{:}\, \mathcal{B}_{n, i}(\mathcal{T}) \geq 1\}\end{equation*}

to be the number of nodes in the largest branch of a given tree, $\mathcal{T}$ . In [Reference Feng, Su and Hu6], it was shown that

(3) \begin{equation}\lim_{n \rightarrow \infty} \mathrm{P} \left(\nu_n(\mathcal{T}_n) \geq \frac{n}{2} \right) = \ln 2,\end{equation}

when $\mathcal{T}_n$ is a URT on n vertices. The first purpose of this section and the next theorem is to extend the result of [Reference Feng, Su and Hu6] to Hoppe trees, and to provide more details about the asymptotic distribution, via exploiting the relation between Hoppe trees and Hoppe permutations. Further, the result in (3) is now extended to an explicit expression for $\lim_{n \rightarrow \infty} \mathrm{P} \left(\nu_n(\mathcal{T}_n) \geq cn \right)$ for $c \in [1/2,1]$ . Once we have the results for the Hoppe tree case, the coupling construction of Section 2 will also generalize these results to WRTs.

Proof.

1. (i) First, we translate the problem into the random permutation setting. We have

   \begin{equation*}\nu_n(\mathcal{T}_n^{\theta}) =_{\rm d} \max\{i \in [n-1] \,{:}\, C_{n-1,i}(\theta) \geq 1\}\eqqcolon \alpha_n(\theta),\end{equation*}
   where $C_{n-1,i}(\theta)$ is the number of cycles of length i in a $\theta$ -biased Hoppe permutation. In this setting, Lemma 5.7 in [Reference Arratia, Barbour and Tavaré2] shows that $\frac{\alpha_n(\theta)}{n}$ converges in distribution to a random variable $\alpha$ with cumulative distribution function
   \begin{equation*}F_{\theta}(x) = 1 + \sum_{k=1}^{\infty} \frac{(- \theta)^k}{k!} \int \cdots \int_{\mathcal{S}_k(x)} \frac{\left(1 - \sum_{j=1}^k y_j \right)^{\theta - 1}}{y_1 \cdots y_k} \, {\rm d} y_1 \cdots {\rm d} y_k, \quad x > 0,\end{equation*}
   where $\gamma$ is Euler’s constant [Reference Finch7] and
   \begin{equation*}\mathcal{S}_k(x)= \bigg\{y_1 >x, \ldots, y_k > x , \sum_{j=1}^k y_j < 1\bigg\}.\end{equation*}
   This proves the first part.

2. (ii) Setting $\theta =1$ in the argument of (i), and recalling that the random permutation in this case reduces to a uniformly random permutation, yields that $\frac{\nu_n(\mathcal{T}_n)}{n}$ converges weakly to a random variable $\nu$ whose cumulative distribution function is given by

   \begin{equation*}F_1(x) =\begin{cases}0, & \text{if } x<0 , \\[3pt]1 + \sum_{k=1}^{\infty} \frac{(-1)^k}{k!} \int \cdots \int_{\mathcal{S}_k(x)} \frac{{\rm d} y_1\ldots {\rm d} y_k}{y_1\ldots y_k}, & \text{if } x \in [0,1] , \\[3pt]1, & \text{if } x >1,\end{cases}\end{equation*}
   where $\mathcal{S}_k(x)$ is as before.

Note that if $x \geq \frac{1}{2}$ , $\mathcal{S}_k(x)= \emptyset$ for any $k \geq 2$ . Thus the expression simplifies for $\frac{1}{2} \leq x \leq 1$ to

\begin{equation*}F_1(x) = 1-\int_{x}^{1}\frac{1}{y_1} \, {\rm d} y_1.\end{equation*}

This in particular implies that the derivative of $F_1(x)$ over $[1/2, 1]$ simplifies to

\begin{equation*}f_1(x) = \frac{1}{x}.\end{equation*}

Hence, for any $c \in [\frac{1}{2}, 1]$ ,

\begin{equation*} \lim_{n \rightarrow \infty} \mathrm{P}(\nu_n(\mathcal{T}_n) \geq c n) = \mathrm{P}(\nu \geq c) = \int_{c}^1 \frac{1}{x} \, {\rm d} x = \ln (1/c).\end{equation*}

Finally, we have

\begin{equation*}\mathrm{E}[\nu] \geq \int_{1/2}^1 x \frac{1}{x} \, {\rm d} x = \frac{1}{2}. \tag*{\ensuremath{\square}}\end{equation*}

Now, let $\mathcal{T}_n^{w}$ be a WRT of size n with weight sequence $(w_i)_{i \in \mathbb{N}}$ such that there is a $k \in \mathbb{N}$ such that, for all $i>k$ , we have $w_i = 1$ . Since we can find a coupling of a Hoppe tree to a WRT in which the number of nodes in the largest branch differs at most by k, the following now follows immediately.

Theorem 6. Let $\mathcal{T}^{w}_{n}$ be a WRT of size n with weight sequence $(w_i)_{i \in \mathbb{N}}$ such that there is a $k \in \mathbb{N}$ such that, for all $i>k$ , we have $w_i = 1$ . Then, for any $c \ in \big[\frac{1}{2}, 1\big]$ , we have

\begin{equation*}\lim_{n \rightarrow \infty} \mathrm{P}(\nu_n(\mathcal{T}_n^w) \leq c n) = 1 - \ln (c^{-1}).\end{equation*}