Count the number of islands in a grid of land and water cells.

Interview question: Number of Islands

Given an `m x n` 2D binary grid `grid` which represents a map of `'1'`s (land) and `'0'`s (water), return the number of islands.

An island is surrounded by water and is formed by connecting adjacent lands horizontally or vertically. You may assume all four edges of the grid are surrounded by water.

## Example
```
Input: grid = [
  ["1", "1", "0", "0", "0"],
  ["1", "1", "0", "0", "0"],
  ["0", "0", "1", "0", "0"],
  ["0", "0", "0", "1", "1"]
]
Output: 3
```

- Each `1` cell is land and each `0` cell is water.
- Land cells belong to the same island if they touch horizontally or vertically.
- Diagonal contact does not connect islands.
- We need to count connected groups of land.

Understand the Grid

- Counting every land cell would overcount large islands.
- Once we discover one land cell, we must find all land connected to it.
- That whole connected component should add only one to the answer.

Naive Counting Is Not Enough

- Scan every cell in the grid.
- When an unvisited land cell is found, it starts a new island.
- Run DFS from that cell to mark every connected land cell as visited.
- Using a visited set avoids mutating the input grid.

Use DFS With a Visited Set

- Only increment the island count when the scan finds land that has not been visited.
- The DFS then visits the entire island, so future cells from the same island are skipped.

Count Island Starts

- Handle an empty grid before reading dimensions.
- Each cell is visited at most once.
- Time complexity is O(rows × cols).
- Space complexity is O(rows × cols) for the visited set and recursion stack in the worst case.