The Maximum (Minimum) Leaf Spanning Tree problem asks for a spanning tree with the largest (smallest) number of leaves. As spanning trees are often computed using graph search algorithms, it is natural to restrict this problem to the set of search trees of some particular graph search, e.g., find the Breadth-First Search (BFS) tree with the largest number of leaves. We study this problem for Generic Search (GS), BFS and Lexicographic Breadth-First Search (LBFS) using search trees that connect each vertex to its first neighbor in the search order (first-in trees) just like the classic BFS tree. In particular, we analyze the complexity of these problems, both in the classical and in the parameterized sense. Among other results, we show that the minimum and maximum leaf problems are in FPT for the first-in trees of GS, BFS and LBFS when parameterized by the number of leaves in the tree. However, when these problems are parameterized by the number of internal vertices of the tree, they are W[1]-hard for the first-in trees of GS, BFS and LBFS.

翻译：最大（最小）叶子生成树问题要求生成一棵具有最多（最少）叶子数的生成树。由于生成树通常使用图搜索算法计算，因此自然地将该问题限制为特定图搜索的搜索树集合，例如，寻找具有最多叶子数的广度优先搜索树。我们研究了泛化搜索、广度优先搜索和字典序广度优先搜索的这一问题，使用将每个顶点连接到搜索顺序中其第一个邻居的搜索树（首入树），类似于经典的广度优先搜索树。具体而言，我们从经典意义和参数化意义两个角度分析了这些问题的复杂性。在其他结果中，我们证明了当以树的叶子数为参数时，针对泛化搜索、广度优先搜索和字典序广度优先搜索的首入树，最小和最大叶子问题属于固定参数可解类。然而，当这些问题以树的内部顶点数为参数时，针对泛化搜索、广度优先搜索和字典序广度优先搜索的首入树，它们是W[1]-难的。