Let B be a standard one-dimensional Brownian motion started at 0. Let L-t,L-v(\B\) be the occupation density of \B\ at level v up to time t. The distribution of the process of local times (L-t,L-v(\B\), v greater than or equal to 0) conditionally given B-t = 0 and L-t,L-0(\B\) = l is shown to be that of the unique strong solution X of the Ito SDE, dX(v) = {4 - X-v(2)(t - integral(0)(v) X-u du)(-1)}dv + 2 root X-v dB(v)on the interval [0, V-t(X)) where V-t(X) := inf{v: integral(0)(v)X(u) du = t), and X-v = 0 for all v greater than or equal to V-t(X). This conditioned form of the Ray-Knight description of Brownian local times arises from study of the asymptotic distribution as n --> infinity and 2k/root n --> l of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements, as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points. The SDE is the continuous analog of a simple description of a Galton-Watson branching process conditioned on its total progeny. For l = 0, corresponding to asymptotics of a uniform random tree, the SDE gives a description of the process of local times of a Brownian excursion which is equivalent to Jeulin's description of these local times as a time change of twice a Brownian excursion. Another corollary is the Biane-Yor description of the local times of a reflecting Brownian ridge as a time-changed reversal of twice a Brownian meander of the same length.
