Approach #1: DepthFirst Search (Recursive) [Accepted]
Intuition and Algorithm
We want to know the area of each connected shape in the grid, then take the maximum of these.
If we are on a land square and explore every square connected to it 4directionally (and recursively squares connected to those squares, and so on), then the total number of squares explored will be the area of that connected shape.
To ensure we don't count squares in a shape more than once, let's use seen
to keep track of squares we haven't visited before. It will also prevent us from counting the same shape more than once.
Complexity Analysis

Time Complexity: , where is the number of rows in the given
grid
, and is the number of columns. We visit every square once. 
Space complexity: , the space used by
seen
to keep track of visited squares, and the space used by the call stack during our recursion.
Approach #2: DepthFirst Search (Iterative) [Accepted]
Intuition and Algorithm
We can try the same approach using a stack based, (or "iterative") depthfirst search.
Here, seen
will represent squares that have either been visited or are added to our list of squares to visit (stack
). For every starting land square that hasn't been visited, we will explore 4directionally around it, adding land squares that haven't been added to seen
to our stack
.
On the side, we'll keep a count shape
of the total number of squares seen during the exploration of this shape. We'll want the running max of these counts.
Complexity Analysis

Time Complexity: , where is the number of rows in the given
grid
, and is the number of columns. We visit every square once. 
Space complexity: , the space used by
seen
to keep track of visited squares, and the space used bystack
.
Analysis written by: @awice