Thank you that worked. See below.

g2 = Graph[g0, VertexShapeFunction -> vsf[shape1, 44, "Arial"], 
   ImageSize -> {3840, 2160}, PerformanceGoal -> "Quality", 
   EdgeStyle -> 
    Black(*edge color*)]; (*This changes the style of the graph*)
yPositions = GraphEmbedding[g2][[All, 2]] // DeleteDuplicates;
prolog = 
  MapThread[{Style[Text["Level " <> ToString@#2, {-.4, #1 + .1}], 
      FontSize -> 44(*font size of the Level *)], Dashing[{.02, .01}],
      Thickness[.001], 
     InfiniteLine[{{0, #1}, {1, #1}}],(*font size of the letters \
inside the circles*)} &, {ReverseSort@yPositions, 
    Range@Length@yPositions}];

GraphPlot[g2, PlotRangePadding -> {Scaled[.05], 0}, Prolog -> prolog]

Thank you very much for working on this. I finally had some time to come back and look at your response. This is exactly what I needed. The final product needs to be a fairly large image. I was able to edit what you posted to increase the image and font size, but I can't seem to figure out how to increase the thickness of the dashed line. How can that be done to make everything look proportionate?

g2 = Graph[g0, VertexShapeFunction -> vsf[shape1, 44, "Arial"], 
   ImageSize -> {3840, 2160}, PerformanceGoal -> "Quality", 
   EdgeStyle -> 
    Black(*edge color*)]; 
yPositions = GraphEmbedding[g2][[All, 2]] // DeleteDuplicates;
prolog = 
  MapThread[{Style[Text["Level " <> ToString@#2, {-.4, #1 + .1}], 
      FontSize -> 44(*font size of the Level *)], Dashed, 
     InfiniteLine[{{0, #1}, {1, #1}}],(*font size of the letters \
inside the circles*)} &, {ReverseSort@yPositions, 
    Range@Length@yPositions}];

GraphPlot[g2, PlotRangePadding -> {Scaled[.05], 0}, Prolog -> prolog]

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

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]

Hello, Thank you! I was able to get a much better image. The function GraphQ[] also is saying it's a valid graph object. It's now a lot easier to visualize. Can I now use g4 and g3 with all the functionality that is given in the Graph documentation?

In[7]:= g = gameTree4["Visualization"]; 
g4 = Graph[EdgeList[g] /. ((VertexShapeFunction /.Options[g, VertexShapeFunction])[[1]])[[1, 1]], VertexLabels -> "Name", ImageSize -> 2000];

In[9]:= GraphQ[g4]

Out[9]= True

In[12]:= g = gameTree4["Visualization"]; 
g3 = Graph[EdgeList[g] /. ((VertexShapeFunction /.Options[g, VertexShapeFunction])[[1]])[[1, 1]], VertexLabels -> "Name", ImageSize -> 2000];

In[14]:= GraphQ[g3]

Out[14]= True

Here is the n = 3

Here is the n = 4

Would this do the job, even if it's a bit hacky and no longer looks much like a tree?

g = gameTree3["Visualization"];
Graph[EdgeList[
   g] /. ((VertexShapeFunction /. 
       Options[g, VertexShapeFunction])[[1]])[[1, 1]], 
 VertexLabels -> "Name", ImageSize -> 2000]

It's basically extracting the edges and vertex labels from your binary tree, then recreating the Graph with vertex labels displayed and all of it enlarged for easier reading.

Side note: InputForm of g helped me figure out what to look for.

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?