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We consider an ordinal tree $T$ on $n$ nodes, with each node assigned a  $d$-dimensional weight vector $\pnt{w} \in \{1,2,\ldots,n\}^d,$ where $d \in\mathbb{N}$ is a constant.  We study path queries as generalizations of well-known {\textit{orthogonalrange queries}}, with one of the dimensions being tree topology rather than alinear order. Since in our definitions $d$ only represents the number ofdimensions of the weight vector without taking the tree topology into account,a path query in a tree with $d$-dimensional weight vectors generalize thecorresponding $(d+1)$-dimensional orthogonal range query.  We solve {\textit{ancestor dominance reporting}} problem as a directgeneralization of dominance reporting problem, %in time $\O((\lg^{d-1}n)/(\lg\lg n)^{d-2}+k)$ in time $\O(\lg^{d-1}{n}+k)$ %and space of $\O(n(\lgn)^{d-1}/(\lg \lg n)^{d-2})$ words, and space of $\O(n\lg^{d-2}n)$ words, where$k$ is the size of the output, for $d \geq 2.$  We also achieve a tradeoff of $\O(n\lg^{d-2+\eps}{n})$ words of space, withquery time of $\O((\lg^{d-1} n)/(\lg\lg n)^{d-2}+k),$ for the same problem,when $d \geq 3.$  We solve {\textit{path successor problem}} in $\O(n\lg^{d-1}{n})$ words ofspace and time $\O(\lg^{d-1+\eps}{n})$ for $d \geq 1$ and an arbitrary constant$\eps > 0.$ We propose a solution to {\textit{path counting problem}}, with$\O(n(\lg{n}/\lg\lg{n})^{d-1})$ words of space and $\O((\lg{n}/\lg\lg{n})^{d})$query time, for $d \geq 1.$  Finally, we solve {\textit{path reporting problem}} in$\O(n\lg^{d-1+\eps}{n})$ words of space and$\O((\lg^{d-1}{n})/(\lg\lg{n})^{d-2}+k)$ query time, for $d \geq 2.$  These results match or nearly match the best tradeoffs of the respectiverange queries. We are also the first to solve path successor even for $d = 1$.

Path and Ancestor Queries on Trees with Multidimensional Weight Vectors