A self-similar invariance of critical binary Galton-Watson trees

A coding of hierarchical structure in finite directed tree graphs was introduced by Robert Horton in 1945; after a modification by Strahler in 1952, this has become a standard river network code defined as follows. Edges along a path adjoining a source (valence 1 vertex) to a junction (valence 3 or higher vertex) are coded as order 1. Such a path of order 1 edges is also referred to as an order 1 stream. Having defined edges and streams of order i, one now recursively defines edges and streams of order i + 1 by the rule that the 'order 1' streams of the tree obtained by pruning the streams of order i (and lower) are assigned order i + 1. This code has helped to identify a number of naturally occurring patterns in river network structure as well as in other naturally occurring dendritic structures. Let T-i,T-j(s) denote the number of order j