The Sackin index and depth of leaves in generalized Schröder trees

Abstract.

Schröder trees are biological models of evolution, with internal nodes having two or three children. We generalize the model to grow from an arbitrary stochastic process of independent nonnegative integers (not necessarily identically distributed). We call such a process the building sequence. We study the depth of leaves and the Sackin index for some specific building sequences, such as constant additions, Bernoulli, and Poisson-like models. We include an example that shows that the methods can be extended to exchangeable sequences.

Keywords:

• Exchangeability
• external path
• martingale
• phylogenesis
• random Schröder trees
• Sackin index

AMS subject classifications;:

• Primary: 90B15
• 05C12
• Secondary: 60C05
• 92B05
• 60F05

Disclosure statement

No potential conflict of interest was reoprted by the author(s).

Notes

1. In the caption of the figure, we use the notation $\stackrel{D}{=}$ to mean equality in law.

2. The ordinary harmonic number H_n is H_n(0), and is often written without the 0 argument.

3. The hypergeometric function on m numerator factors and n denominator factors is ${}_m{F}_n(a_1,a_2,\dots ,a_m,b_1,b_2,\dots b_n;z)={\sum }_{k=0}^{\mathrm{\infty }}\frac{(a_1{)}_k(a_2{)}_k\dots (a_m{)}_k}{(b_1{)}_k(b_2{)}_k\dots (b_n{)}_k}\times \frac{z^k}{k!},$ where $(x{)}_k=x(x+1)\cdots (x+k-1)$ is Pochhammer’s symbol for the k-times rising factorial of x∈ℝ, with $k\in {\mathbb{Z}}^{+}$. Some sources use the notation $[\genfrac{}{}{0ex}{}{a_1,\dots ,a_m}{b_1,\dots ,b_n}|z]$ for ${}_m{F}_n(a_1,a_2,\dots ,a_m,b_1,b_2,\dots b_n;z)$; see Graham et al.[7], for example. We prefer the visually pleasing double-decker notation in displays, as it is easier to parse. We only use the notation ${}_m{F}_n$ in the text to conserve space, when no arguments are specified.