It is shown that if $\kappa$ is an uncountable successor cardinal in $L\lbrack 0^\sharp\rbrack$, then there is a normal tree $\mathbf{T} \in L \lbrack 0^\sharp\rbrack$ of height $\kappa$ such that $0^\sharp \not\in L\lbrack\mathbf{T}\rbrack$. Yet $\mathbf{T}$ is $<\kappa$-distributive in $L\lbrack 0^\sharp\rbrack$. A proper class version of this theorem yields an analogous $L\lbrack 0^\sharp\rbrack$-definable tree such that distinct branches in the presence of $0^\sharp$ collapse the universe. A heretofore unutilized method for constructing in $L\lbrack 0^\sharp\rbrack$ generic objects for certain $L$-definable forcings and "exotic sequences", combinatorial principles introduced by C. Gray, are used in constructing these trees.