Another way to approach the linear combination is to look at it geometrically. This is where the idea of barycentric coordinates can help. A barycentric coordinate system describes the location of a point as the weighted sum of the regular coordinates of the vertices forming a simplex. In other words, it describes a linear combination with respect to a set of points, where in -dimensional space.
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The algorithm also visits children in order of distance to the query point. This means it checks the most promising quadrants first, which tends to find a good candidate early and enables aggressive pruning of the remaining quadrants. Without this ordering, the algorithm would still produce the correct result, but it would prune fewer nodes.