Spiderweb Trees

Let's call a graph with $$$n$$$ vertices, each of which has it's own point $$$A_i = (x_i, y_i)$$$ with integer coordinates, a planar tree if:

• All points $$$A_1, A_2, \ldots, A_n$$$ are different and no three points lie on the same line.
• The graph is a tree, i.e. there are exactly $$$n-1$$$ edges there exists a path between any pair of vertices.
• For all pairs of edges $$$(s_1, f_1)$$$ and $$$(s_2, f_2)$$$, such that $$$s_1 \neq s_2, s_1 \neq f_2,$$$ $$$f_1 \neq s_2$$$, and $$$f_1 \neq f_2$$$, the segments $$$A_{s_1} A_{f_1}$$$ and $$$A_{s_2} A_{f_2}$$$ don't intersect.

Imagine a planar tree with $$$n$$$ vertices. Consider the convex hull of points $$$A_1, A_2, \ldots, A_n$$$. Let's call this tree a spiderweb tree if for all $$$1 \leq i \leq n$$$ the following statements are true:

• All leaves (vertices with degree $$$\leq 1$$$) of the tree lie on the border of the convex hull.
• All vertices on the border of the convex hull are leaves.

An example of a spiderweb tree:

The points $$$A_3, A_6, A_7, A_4$$$ lie on the convex hull and the leaf vertices of the tree are $$$3, 6, 7, 4$$$.

Refer to the notes for more examples.

Let's call the subset $$$S \subset \{1, 2, \ldots, n\}$$$ of vertices a subtree of the tree if for all pairs of vertices in $$$S$$$, there exists a path that contains only vertices from $$$S$$$. Note that any subtree of the planar tree is a planar tree.

You are given a planar tree with $$$n$$$ vertexes. Let's call a partition of the set $$$\{1, 2, \ldots, n\}$$$ into non-empty subsets $$$A_1, A_2, \ldots, A_k$$$ (i.e. $$$A_i \cap A_j = \emptyset$$$ for all $$$1 \leq i < j \leq k$$$ and $$$A_1 \cup A_2 \cup \ldots \cup A_k = \{1, 2, \ldots, n\}$$$) good if for all $$$1 \leq i \leq k$$$, the subtree $$$A_i$$$ is a spiderweb tree. Two partitions are different if there exists some set that lies in one parition, but not the other.

Find the number of good partitions. Since this number can be very large, find it modulo $$$998\,244\,353$$$.