## A self-similar invariance of critical binary Galton-Watson trees.(English)Zbl 0967.60081

A pruning and compression operator on finite trees is introduced. The pruning cuts away all branches leading to only one leaf; the compression identifies all nodes with only one daughter node on a branch. A distribution, $$P$$, on finite trees is called stochastically self-similar if, conditional on the tree not being just the root, the distribution of the pruned and compressed tree is also $$P$$. The operator can be applied repeatedly, until only the root is left. One plus the number of applications needed for this is called the order of the tree, denoted by $$W$$. The order of any node is the order of the sub-tree emanating from it. A stream of order $$i$$ is that part of a line of descent containing only nodes of order $$i$$; the terminal node of a stream is the one that has no daughter nodes of order $$i$$. Let $$T_{i,j}$$ be the number of trees of order $$j$$ $$(<i)$$ emanating from the non-terminal nodes of a stream of order $$i$$. Consideration of $$T_{i,j}$$ is motivated by empirical observations on the branching structure of river systems. The main result is that if $$P$$ is a Galton-Watson law with finite maximum family size, the following are equivalent: $$P$$ is stochastically self-similar; the family size is critical binary splitting; $$W$$ is geometric; $$E[T_{i,j}\mid W \geq i+1]$$ is a function of $$i-j$$ only.

### MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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