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We consider uniform spanning tree (UST) in topological polygons with $2N$marked points on the boundary with alternating boundary conditions. In [LPW21],the authors derive the scaling limit of the Peano curve in the UST. They arevariants of SLE$_8$. In this article, we derive the scaling limit of theloop-erased random walk branch (LERW) in the UST. They are variants of SLE$_2$.The conclusion is a generalization of [HLW20,Theorem 1.6] where the authorsderive the scaling limit of the LERW branch of UST when $N=2$. When $N=2$, thelimiting law is SLE$_2(-1,-1; -1, -1)$. However, the limiting law is nolongerin the family of SLE$_2(\rho)$ process as long as $N\ge 3$.

Loop-Erased Random Walk Branch of Uniform Spanning Tree in Topological Polygons