Constructing a tree from an array for a DFS solver

I am making a maze solver using DFS and I want to implement the search tree for it but I am a bit new to artificial intelligence and I want a little bit of help on this matter.

Let me give a maze example first:

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char maze[5][9] = 
    "#########",
    "#  #    #",
    "# ##  # #",
    "#     # #",
    "#########",

So my search tree for DFS should be something like this:
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    "#########",
    "#12#15 10 11 #",
    "#3##14 9 #12 #",
    "#456 7 8 #13 #",
    "#########",

1st child of parent -> Right Cell if it is empty

2nd child of parent -> Bottom Cell if it is empty

3rd child of parent -> Left Cell if it is empty

4th child of parent -> Top Cell if it is empty


My solver will receive as an argument my maze array. My questions are these:

1st question: Is this actually the way the nodes are going to get visited by my actor?


2nd question: In the code do I need to declare 15 as child of 10? (also in other cases like 9 with 14)


3rd question: When my solver receives the array do I need to make a pre-process on the array and construct the tree from the array or will my actor construct it as he goes?
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