Continuum tree limit for the range of random walks on regular trees

Let b be an integer greater than I and let W-epsilon = (W-n(epsilon); n > 0) be a random walk on the b-ary rooted tree U-b, starting at the root, going up (resp. down) with probability 1/2 + epsilon (resp. 1/2 - epsilon), epsilon is an element of (0, 1/2), and choosing direction i epsilon {1,...,b) when going up with probability a(i). Here a = (a(1),...,a(b)) stands for some nondegenerated fixed set of weights. We consider the range {W-n(epsilon); n >= 0} that is a subtree Of U-b. It corresponds to a unique random rooted ordered tree that we denote by tau(epsilon). We rescale the edges of tau(epsilon) by a factor and we let e go to 0: we prove that correlations due to frequent backtracking of the random walk only give rise to a deterministic phenomenon taken into account by a positive factor gamma (a). More precisely, we prove that tau(epsilon) converges to a continuum random tree encoded by two independent Brownian motions with drift conditioned to stay positive and scaled in time by gamma (a). We actually state the result in the more general case of a random walk on a tree with an infinite number of branches at each node (b = infinity) and for a general set of weights a = (a(n), n > 0).