1., 2., and 3. can be accomplished using a VertexShapeFunction. I provided an example of this in an earlier answer. These requirements need a different implementation.

The helper function I provided in an earlier answer

ClearAll[visualizationToGraph];
visualizationToGraph[viz_, opts : OptionsPattern[Graph]] :=
  Graph[EdgeList[viz] /. ((VertexShapeFunction /. 
                Options[viz, VertexShapeFunction])[[1]])[[1, 1]], opts]

g = gameTree2["Visualization"];
g0 = visualizationToGraph[g];

VertexShapeFunction.

vsf[shape_, fontSize_, fontFamily_][position_, name_, dimensions_] :=
{EdgeForm[Black],
 White,
 shape[position],
 Text[Style[name, fontSize, Black, FontFamily -> fontFamily], position]};

shape is a function that takes a position and returns a Graphics primitive located at that position. E.g.

shape1 = Disk[#1, .2] &;
shape2 = RegularPolygon[#, .25, 6] &;

Using shape2

g1 = Graph[g0,
  VertexShapeFunction -> vsf[shape2, 10, "Arial"],
  ImageSize -> {1000, 500},
  PerformanceGoal -> "Quality"]

The requirements for 4. are not very clear. Here is one interpretation

g2 = Graph[g0,
   VertexShapeFunction -> vsf[shape1, 10, "Arial"],
   ImageSize -> {1000, 500},
   PerformanceGoal -> "Quality"];
yPositions = GraphEmbedding[g2][[All, 2]] // DeleteDuplicates;
prolog = 
  MapThread[{Text["Level " <> ToString@#2, {-.4, #1 + .1}], Dashed, InfiniteLine[{{0, #1}, {1, #1}}]} &,
   {ReverseSort@yPositions, Range@Length@yPositions}];

GraphPlot[g2,
   PlotRangePadding -> {Scaled[.05], 0},
   Prolog -> prolog] // 
 Labeled[#, Style["Game Tree 3", 14, Black, Bold], Top] &

You can add e.g. Thickness[0.2] before Dashed, however, this just increases the vertical size of the dashes which may not look good. You will have to experiment with a combination of Dashing (rather than Dashed) and Thickness to make it to look right.. e.g. Dashing[{.02, .02}], Thickness[.02]

Hello,

I've attached a notebook file with what I'm currently trying to do, and thank you all for putting together these response's. They have been tremendously helpful. I've moved onto playing around with a simpler graph using these functions. I need to change a few things to conform with the style of my document. I was able to increase the text size and play around with the positioning of the labels; however, I'm having trouble with some other changes.

I'm trying to do numerous things.

1. I only want to see the black outline of the circle without any fill.
2. I also need the text to be in Arial font. (Tried using LabelStyle -> {FontFamily -> "Arial"} but it doesn't appear to be working.
3. How do I change the size and/or shape of the vertices?
4. Is there anyway to add horizontal lines that would indicate the height of the tree? (With some simple label centered in-between the horizontal lines on the left or right.)

Help with any of this would be much appreciated!

Thank you

Hi Peter,

Nice!. To make it look a little more like Ziggy's plot

vsf[width_, fontSize_][position_, name_, dimensions_] := 
  Inset[Framed[
    Pane[Style[name, fontSize, Black], width, Alignment -> Center],
    Background -> LightYellow,
    RoundingRadius -> 5], position];

g3g = Graph[
   DirectedEdge @@@ EdgeList[g] /. ((VertexShapeFunction /. Options[g, VertexShapeFunction])[[1]])[[1, 1]],
   VertexLabels -> "Name",
   ImageSize -> 2000];

Graph[g3g,
 VertexShapeFunction -> vsf[25, 8],
 VertexLabels -> None,
 EdgeStyle -> Arrowheads[.005],
 GraphLayout -> {"LayeredEmbedding", "RootVertex" -> "0", "LeafDistance" -> .4},
 PerformanceGoal -> "Quality"]

Have to experiment with the values passed to vsf, GraphLayout, AspectRatio, ImageSize to get a better separation between the vertices.

Can you recommend any of Graphs built in functions to find out more information about the tree?

There are several Graph algorithms that you could use depending on what question(s) you are trying to answer. Take a look at this guide page. Since the graph is a tree it can be converted to a Tree using GraphTree.

g4 = gameTree4["Visualization"];
g4g = visualizationToGraph[g4];
g4t = Graph[DirectedEdge @@@ EdgeList[g4g]] // GraphTree[#, "0"] &;

TreeLeafCount@g4t
(* 3941 *)

TreeCount[g4t, _?(StringStartsQ[#, "0111"] &)]
(* 1121 *)

TreeDepth[g4t]
(* 19 *)

TreeDepth[g4t, "0111101100100"]
(* 12 *)

Take a look at this guide page for details.

Can I now use g4 and g3 with all the functionality that is given in the Graph documentation?

Yes, you can. Here is a little helper function that encapsulates extracting the graph data from the visualization (from Peter's answer) and constructing a usable Graph from it. It accepts all of the options that Graph supports. E.g.

ClearAll[visualizationToGraph];
visualizationToGraph[viz_, opts : OptionsPattern[Graph]] :=
 Graph[EdgeList[viz] /. ((VertexShapeFunction /. 
        Options[viz, VertexShapeFunction])[[1]])[[1, 1]], opts]

g4 = gameTree4["Visualization"];
visualizationToGraph[g4,
 GraphLayout -> "SpringElectricalEmbedding",
 VertexLabels -> Placed["Name", Tooltip], (* Use a Tooltip to display the vertex name *)
 VertexStyle -> Red,
 ImageSize -> 1000]

Hi Ziggy

It seems that the "Visualization" function is spitting out a graph data type, but I'm unable to use the graph options to make my vertices larger

The reason for this is that the Graph returned does not have the usual structure, take a look at its InputForm. The VertexShapeFunction refers to the undocumented DataStructure`Visualization`RenderVertex function. That can be overridden by e.g.

Graph[gameTree3["Visualization"], VertexShapeFunction -> "Circle"]

But the data associated with each node is lost. Since Graph is an Atom there is no easy way to extract the data.

Is there a reason for using DataStructure rather than Graph to build the tree?