How much branching complexity should we expect in a random tree of a given size? Counting leaves is not enough: a long chain of side branches and a balanced tree can have very different branching orders.

This candidate studies two mathematical models. They share the language of branching, but the results and limits are different.

In critical Tokunaga trees, the paper conditions on exactly a given number of leaves. It derives not just the leading logarithmic growth of the branching order, but a distribution describing its fluctuations. The correction depends on the fractional part of a logarithm, producing a repeating dependence on scale. The familiar binary-tree case was already classical; the claim covers the full stated parameter family.

For Kingman’s coalescent, earlier work established the leading exponential decay of branch frequencies. This paper adds a positive prefactor. That is stronger than knowing an exponential rate: after the exponential decay is removed, the normalized frequency approaches a finite, nonzero number.

The proof needs an extra structural bound. Converging ratios alone would not establish that positive limit. A fractional-linear comparison supplies the missing control.

The package contains written analytical proofs, exact finite algebraic checks and numerical diagnostics. A numerical instability identified by the supplied review has been repaired and tested at two resolutions. Those calculations are not certified decimal expansions and do not replace the proofs.

The Kingman result takes the large-tree limit first, at each fixed order. It does not allow arbitrary simultaneous limits. Nor does either theorem establish that real river networks follow these models.

This is Evidence Press, Beyond the branching exponent, dated the fifth of October twenty twenty-six. The paper and evidence package are linked on the release page. The work remains an unrefereed candidate. This synthetic AI voice is an explanation, not additional scientific evidence.