For each tree, pick the smallest relative distance between the woodcutter by using a modified dy (assuming the major axis of your ellipse is on the x axis).
Modified dy = (Woodcutter.Y - Tree.Y) * Ellipse.Width / Ellipse.Height
Distance = (dx^2 + dy^2)^-2
You could even just pick the smallest of the Abs(dx) + Abs(dy), since the actual distance value is not really needed.
A function returning the UID of the closest tree would work great !