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In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved. Itstates that symmetric processes associated with resistance metric measurespaces converge when the underlying spaces converge with respect to theGromov-Hausdorff-vague topology, and a certain uniform recurrence condition issatisfied. Such a theorem finds particularly nice applications if theresistance metric measure space is a metric measure tree. To illustrate this,we state functional limit theorems in old and new examples of suitably rescaledrandom walks in random environment on trees. First, we take a criticalGalton-Watson tree conditioned on its total progeny and a non-lattice branchingrandom walk on $\mathbb{R}^d$ indexed by it. Then, conditional on that, weconsider a biased random walk on the range of the preceding. Here, bynon-lattice we mean that distinct branches of the tree do not intersect oncemapped in $\mathbb{R}^d$. This excludes the possibility that the random walk onthe range may jump from one branch to the other without returning to the recentcommon ancestor. We prove, after introducing the bias parameter$\beta^{n^{-1/4}}$, for some $\beta>1$, that the biased random walk on therange of a large critical non-lattice branching random walk converges to aBrownian motion in a random Gaussian potential on Aldous' continuum random tree(CRT). Our second new result introduces the scaling limit of theedge-reinforced random walk on a size-conditioned Galton-Watson tree withfinite variance as a Brownian motion in a random Gaussian potential on the CRTwith a drift proportional to the distance to the root.

Invariance principles for random walks in random environment on trees