Graphviz non-overlapping tree - graphviz

Question

I am trying to use Graphviz to produce a tree such as the following:
I have it almost working, as shown below:
My problems are the following:
Subtrees should never overlap, but the PP tree and the (.) tree overlap.
Subtrees should always be fairly regular, so that if there are multiple children, the edges should go to the left and right of the parent. Again, this does not work properly for the PP tree.
I have fixed (1) by using subgraph clusters, however, this introduces a few new issues. Namely, the edges are not straight, and I can't figure out how to hide the borders without leaving tons of empty space.
Is there a better way to force the tree formatting? It feels like it shouldn't be too unusual.
My code is below (generated by Python, sorry for unhelpful names):
graph {
subgraph 0 {
subgraph 1 {
0 [label=ROOT group=0 shape=plain]
subgraph 8 {
1 [label=S group=1 shape=plain]
subgraph 11 {
2 [label=NP group=2 shape=plain]
subgraph 15 {
3 [label=PRP group=3 shape=plain]
4 [label=I shape=plain]
3 -- 4
4 [label=I group=3]
}
2 -- 3
3 [label=PRP group=2]
}
subgraph 24 {
5 [label=VP group=5 shape=plain]
subgraph 28 {
6 [label=VBD group=6 shape=plain]
7 [label=solved shape=plain]
6 -- 7
7 [label=solved group=6]
}
subgraph 41 {
8 [label=NP group=8 shape=plain]
subgraph 45 {
9 [label=DT group=9 shape=plain]
10 [label=the shape=plain]
9 -- 10
10 [label=the group=9]
}
subgraph 63 {
11 [label=NN group=11 shape=plain]
12 [label=problem shape=plain]
11 -- 12
12 [label=problem group=11]
}
8 -- 9
8 -- 11
}
subgraph 77 {
13 [label=PP group=13 shape=plain]
subgraph 81 {
14 [label=IN group=14 shape=plain]
15 [label=on shape=plain]
14 -- 15
15 [label=on group=14]
}
subgraph 89 {
16 [label=NP group=16 shape=plain]
subgraph 93 {
17 [label=DT group=17 shape=plain]
18 [label=the shape=plain]
17 -- 18
18 [label=the group=17]
}
subgraph 102 {
19 [label=NN group=19 shape=plain]
20 [label=bus shape=plain]
19 -- 20
20 [label=bus group=19]
}
16 -- 17
16 -- 19
}
13 -- 14
13 -- 16
}
5 -- 6
5 -- 8
5 -- 13
}
subgraph 114 {
21 [label="." group=21 shape=plain]
22 [label="." shape=plain]
21 -- 22
22 [label="." group=21]
}
1 -- 2
1 -- 5
1 -- 21
}
0 -- 1
1 [label=S group=0]
}
}
ranksep=0.2
}

Answer 1

You are correct, Graphviz has "problems" with trees. That said, here is your graph, with clusters (peripheries=0), splines=false, and margin=2:
graph {
graph [splines=false]
subgraph cluster_0 {
peripheries=0
margin=2
subgraph cluster_1 {
0 [label=ROOT group=0 shape=plain]
subgraph cluster_8 {
1 [label=S group=1 shape=plain]
subgraph cluster_11 {
2 [label=NP group=2 shape=plain]
subgraph cluster_15 {
3 [label=PRP group=3 shape=plain]
4 [label=I shape=plain]
3 -- 4
4 [label=I group=3]
}
2 -- 3
3 [label=PRP group=2]
}
subgraph cluster_24 {
5 [label=VP group=5 shape=plain]
subgraph cluster_28 {
6 [label=VBD group=6 shape=plain]
7 [label=solved shape=plain]
6 -- 7
7 [label=solved group=6]
}
subgraph cluster_41 {
8 [label=NP group=8 shape=plain]
subgraph cluster_45 {
9 [label=DT group=9 shape=plain]
10 [label=the shape=plain]
9 -- 10
10 [label=the group=9]
}
subgraph cluster_63 {
11 [label=NN group=11 shape=plain]
12 [label=problem shape=plain]
11 -- 12
12 [label=problem group=11]
}
8 -- 9
8 -- 11
}
subgraph cluster_77 {
13 [label=PP group=13 shape=plain]
subgraph cluster_81 {
14 [label=IN group=14 shape=plain]
15 [label=on shape=plain]
14 -- 15
15 [label=on group=14]
}
subgraph cluster_89 {
16 [label=NP group=16 shape=plain]
subgraph cluster_93 {
17 [label=DT group=17 shape=plain]
18 [label=the shape=plain]
17 -- 18
18 [label=the group=17]
}
subgraph cluster_102 {
19 [label=NN group=19 shape=plain]
20 [label=bus shape=plain]
19 -- 20
20 [label=bus group=19]
}
16 -- 17
16 -- 19
}
13 -- 14
13 -- 16
}
5 -- 6
5 -- 8
5 -- 13
}
subgraph cluster_114 {
21 [label="." group=21 shape=plain]
22 [label="." shape=plain]
21 -- 22
22 [label="." group=21]
}
1 -- 2
1 -- 5
1 -- 21
}
0 -- 1
1 [label=S group=0]
}
}
ranksep=0.2
}
Giving: