Okisaka et al. (2017) investigated the eigen-distribution for multi-branching trees weighted with (a,b) on correlated distributions, which is a weak version of Saks and Wigderson’s (1986) weighted trees. In the present work, we concentrate on the studies of eigen-distribution for multi-branching weighted trees on independent distributions. In particular, we generalize our previous results in Peng et al. (Inform Process Lett 125:41–45, 2017) to weighted trees where the cost of querying each leaf is associated with the leaf and its Boolean value. For a multi-branching weighted tree, we define a directional algorithm and show it is optimal among all the depth-first algorithms with respect to the given independent distribution. For some balanced multi-branching trees weighted with (a,b) on the assumption 0 < r < 1 (r is the probability that the root has value 0), we further prove that if an independent distribution d achieves the distributional complexity, then d turns out to be an independent and identical distribution.

冲坂等。（2017）研究了在相关分布上用（a，b）加权的多分支树的特征分布，这是Saks和Wigderson（1986）加权树的弱版本。在当前的工作中，我们专注于在独立分布上对多分支加权树的特征分布进行研究。特别是，我们在Peng等人中概括了我们以前的结果。（通知Process Lett 125：41–45，2017年）到加权树，其中查询每片叶子的成本与叶子及其布尔值相关。对于多分支加权树，我们定义了一种方向算法，并表明对于给定的独立分布，在所有深度优先算法中它是最佳的。对于一些加权的平衡多分支树a，b）假设0 < r <1（r是根的值为0的概率），我们进一步证明，如果独立分布d达到分布复杂度，则d成为独立且相同的分布。