  We study the inhomogeneous continuum random trees (ICRT) that arise as weak
limits of birthday trees. We give a description of the exploration process, a
function defined on [0,1] that encodes the structure of an ICRT, and also of
its width process, determining the size of layers in order of height. These
processes turn out to be transformations of bridges with exchangeable
increments, which have already appeared in other ICRT related topics such as
stochastic additive coalescence. The results rely on two different
constructions of birthday trees from processes with exchangeable increments, on
weak convergence arguments, and on general theory on continuum random trees.
