Inhomogeneous branching trees with symmetric and asymmetric offspring and their genealogies

TL;DR

The paper develops a comprehensive, non-Markovian framework for inhomogeneous populations by introducing symmetric and asymmetric branching trees and their Sevast'yanov processes, including reduced variants that track extant lineages. It provides a rigorous generating-function approach to characterize the full and reduced processes, establishes a genealogical theory with a novel branching property under an enlarged conditioning, and derives recursive descriptions and simulation schemes for genealogies. The results illuminate how age- and time-dependent reproduction can be treated tractably via integral equations and a structured decomposition, enabling precise distributional results and practical genealogy simulations in non-Markovian settings. Collectively, the framework expands modeling flexibility for CMJ-type processes and their genealogies, with potential applications in infectious disease modeling and phylogenetic inference where age-dependent reproduction plays a key role.

Abstract

We define symmetric and asymmetric branching trees, a class of processes particularly suited for modeling genealogies of inhomogeneous populations where individuals may reproduce throughout life. In this framework, a broad class of Crump-Mode-Jagers processes can be constructed as (a)symmetric Sevast'yanov processes, which count the branches of the tree. Analogous definitions yield reduced (a)symmetric Sevast'yanov processes, which restrict attention to branches that lead to extant progeny. We characterize their laws through generating functions. The genealogy obtained by pruning away branches without extant progeny at a fixed time is shown to satisfy a branching property, which provides distributional characterizations of the genealogy.

Figures (4)

• Figure 1: Symmetric (left) and asymmetric (right) branching trees, explicitly labeled, both have the same Neveu tree and branch lengths. In the symmetric tree, the progeny set $\mathrm{Pr}_{13}$ (green), the ancestor set $\mathrm{An}_{121}$ (yellow), and selected variables are highlighted. In the asymmetric tree, the individual ${12,121,1211}$ (red) and selected variables are highlighted. The asymmetric representation emphasizes that rank-1 branches are continuations of their mother.
• Figure 2: Fundamental decomposition of the asymmetric branching tree from Figure \ref{['fig:sym_asym']} along the line $I={2,12,131}$. The pruned tree $\mathcal{K}_I$ is shown in grey, with the subtrees $\mathcal{T}_2$ (red), $\mathcal{T}_{12}$ (yellow), and $\mathcal{T}_{131}$ (green) highlighted. By Proposition \ref{['thm:StrongBranching']}, these subtrees are independent and each distributed as the whole tree, given the pruned tree. The same decomposition and branching property apply to the symmetric branching tree.
• Figure 3: On the left: The asymmetric tree from Figure \ref{['fig:sym_asym']}, censored at time $T$, with branches without extant progeny shown in grey, the least common ancestor $\lambda^T$ in green, and its offspring with extant progeny, $\Lambda^T$, in yellow. On the right: The associated genealogy with selected variables highlighted. Genealogies are drawn in the style of symmetric trees, even though birth ages may be non-zero; this is done to avoid suggesting that they represent asymmetric trees with linearly increasing birth ages. The same genealogical constructions apply to the symmetric branching tree, correcting for birth ages always being zero.
• Figure 4: The first-generation fundamental decomposition of the genealogy in Figure \ref{['fig:genealogy']} into the root (grey) and the two first-generation sub-genealogies. By Theorem \ref{['thm:lcabranching']}, these subtrees are independent and distributed as the whole genealogy, conditional on the pruned underlying branching tree (not shown; see Figure \ref{['fig:genealogy']}).

Theorems & Definitions (38)

• Proposition 2.1
• Proposition 2.2: Strong branching property
• Lemma 3.1: Principle of first generation
• Lemma 3.2
• Proposition 3.3
• Example 3.5
• Theorem 3.7
• Proposition 3.8
• Corollary 3.9
• Corollary 3.10