Let $B$ be a standard one-dimensional Brownian motion started
at 0. Let $L_{t,v}(|B|)$ be the occupation density of $|B|$ at level $v$
up to time $t$. The distribution of the process of local times
$(L_{t,v}(|B|)v\geq0)$ conditionally given $B_{t}= 0$ and $L_{t,0}(|B|)=l$ is
shown to be that of the unique strong solution $X$ of the Itô SDE,
\[dX_v = \Big\{4 - X^2_v\Big(t - \textstyle\int_0^v X_u\,du\Big)^{-1}\Big\}dv + 2\sqrt{X_v}\,dB_v\]
¶ on the interval $[0,V_{t}(X))$, where $V_{t}(X):=
\inf{v:\int_0^v X_u,du=t}$, and $X_v= 0$ for all $v\geqV_t(X)$. This conditioned
from study of the Ray-Knight description of Brownian local times arises from
study of the asymptotic distribution as $n\rightarrow\infty$ and
$2k\sqrt{n\rightarrowl}$ 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.