4. Branching model and its associated treelike structure

In this section we consider a continuous time binary critical branching process <sup>Z</sup>Kx ð Þ<sup>t</sup> on <sup>D</sup><sup>0</sup> [9], whose branching rate is given by a parameter <sup>λ</sup>j j <sup>x</sup> 2 , whose branching mechanism is binary with equiprobability, and whose descendant branching particle behavior is determined by the kernel Kx (cf. [10]). Next, taking notice of the tree structure which the process <sup>Z</sup>Kx ð Þ<sup>t</sup> determines, we denote the space of marked trees

$$\rho = (t, (t\_m), (\chi\_m), (\eta\_m), m \in \mathcal{V}) \tag{9}$$

by <sup>Ω</sup> (see [11]). We also consider the time-reversed law of <sup>Z</sup>Kx ð Þ<sup>t</sup> being a probability measure on Ω as Pt,x ∈Pð Þ Ω . Here, t denotes the birth time of common ancestor, and the particle xm dies when η<sup>m</sup> ¼ 0, while it generates two descendants xm1, xm<sup>2</sup> when η<sup>m</sup> ¼ 1. On the other hand,

$$\mathcal{V} = \bigcup\_{\ell \ge 0} \{\mathbf{1}, \mathbf{2}\}^{\ell}$$

is a set of all labels, namely, finite sequences of symbols with length ℓ, which describe the whole tree structure given [12]. For ω∈ Ω we denote by N ð Þ ω the totality of nodes being the branching points of tree; let Nþð Þ ω be the set of all nodes m being a member of V \ N ð Þ ω , whose direct predecessor lies in N ð Þ ω and which satisfies the condition tmð Þ ω . 0, and let N�ð Þ ω be the same set as described above but satisfying tmð Þ ω ⩽ 0. Finally, we put

$$N(\boldsymbol{\alpha}) = N\_+(\boldsymbol{\alpha}) \cup N\_-(\boldsymbol{\alpha}).\tag{10}$$
