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A complete binary dendritic tree is considered in which each branch (cylinder) receives random input. The depolarization on each cylinder satisfies a stochastic cable equation. The Laplace transform of the solution can be found by solving a linear system of 2[2n+1-1] equations when there are n orders of branching. The Laplace transform method is shown to yield the exact solution for a single cylinder with white noise current injection. A tree with n branches emanating from a common origin (soma) is then considered. With each branch electrotonic length the same, the depolarization at the soma may be found exactly and expressions may be obtained for its mean and variance.