This paper is devoted to the ongoing study on the equilibrium points of AND-OR trees. Liu and Tanaka (2007, 2007a) characterized the eigen-distributions that achieve the distributional complexity, and among others, they proved the uniqueness of eigen-distribution for a uniform binary tree. Later, Suzuki and Nakamura (2012) showed that the uniqueness fails if only directional algorithms are allowed. Peng et al. (2016) extended the studies on eigen-distributions to balanced multi-branching trees of height 2. But, it remains open whether the uniqueness still holds or not for general multi-branching trees. To this end, we introduce the weighted trees, namely, trees with weighted cost depending on the value of a leaf. Using such models, we prove that for balanced multi-branching trees, the uniqueness of eigen-distribution holds w.r.t. all deterministic algorithms, but fails w.r.t. only directional algorithms.