Proposition: Root-mark counterexample

Path concentration does not imply concentration of one root mark.

Proof. Take $p_0=p_2=1/2$. At every node attach an independent Bernoulli$(1/2)$ mark $U_u$, and define $b(k,i,U)=U$ for $k>0$. Then $0\le b\le k$, so Theorem thm:path applies. For every conditioned size $N>1$, the root is internal and its mark remains Bernoulli$(1/2)$, independently of the size-conditioning event. It does not concentrate on its mean.