# Depth-First Search: Conceptual

High-level, language-agnostic introduction to depth-first search with theoretical and practical discussion.

## Depth-First Search Algorithm

Depth-First Search is an algorithm used for searching tree data structures for a particular node, or node with a particular value associated with it. Depth-First Search is also more generally used as a tree traversal algorithm, specifying an order in which to exhaustively access all nodes of a tree. The algorithm begins at the root node and explores deeper into the tree until it reaches a leaf node. Only then will it back-track up the tree until it finds an unexplored child node. This process continues until the desired node is found, or all nodes have been explored.

There is an intuitive recursive implementation of this algorithm which we will discuss in this article, but this search method can also be implemented with an iterative method that uses a stack to store roots of unexplored subtrees. Consider Breadth-First Search, where we traverse a tree by preferring to explore laterally between siblings before moving to deeper levels. While in Depth-First Search we traverse the tree vertically, from parent to child, before exploring any siblings of that parent. Depth-First Search is an exhaustive search and thus will find the targeted node if it exists in the tree. This, however, has practical implications. If a search tree is very large, or infinite in size, the Depth-First Search algorithm may never halt. ### The Algorithm

#### Recursive Implementation

The recursive version of the algorithm works by starting at the root node, and breaking the tree up into subtrees, until it finds the target node, or until every node in the tree has been considered as the root of a subtree. We recursively call the function on all of our root’s children, treating each child node as a root of its own subtree. We define a function that accepts a tree node and a target value as input parameters. The recursive DFS algorithm implements the following logic:

• If the input node value matches our target value then return the input node.
• For each child of the input node, recursively call this function and return the first non-null value returned by a recursive call.
• If this root node has no children, or the recursive calls did not return any node, then return null.

To search a tree with this function, we invoke the function with the root node of our tree.

#### Iterative Implementation

The iterative algorithm does not make use of any recursive calls. Instead, we maintain a stack of references to unexplored siblings of the nodes we have already accessed. The recursive algorithm is effectively doing something very similar, but the program call stack is implicitly used to store the path from the root to the current node. With the iterative algorithm, we need to implement a stack ourselves. These two implementations have the same time and space complexity, so the choice of which to implement is usually a matter of personal preference.

### Complexity

Depth-First Search has a time complexity of `O(n)` where n is the number of nodes in the tree. In the worst case, we will examine every node of a tree.

Depth-First Search has a space complexity of `O(n)` where n is the number of nodes in the tree. In the worst case, we will need to store a reference to every node in a stack. Consider an adversarial example of a linked list as a type of tree with no branching. Trees like this could reach the worst-case space complexity if the target node is the leaf node or is absent from the tree altogether.

### Practical Considerations

DFS is a popular tree search algorithm for its intuitive and concise implementations. Depth-First search, and other tree traversal algorithms like it, can be found in many modern applications. DFS can be used for searching for objects in a data structure like files in a file system. More abstract applications can be found in domains like artificial intelligence, for example, searching through possible moves in a game like chess, or searching for a path through a maze.

Depth-First Search is an exhaustive method, so consider that some trees may be infinitely large or just large enough that they are practically impossible to completely traverse. In practice, we can limit the depth allowed to search by the algorithm. Consider something like an algorithm for playing chess. We can simulate a series of moves and responses, but the search space may be so large that no modern computer could reasonably explore all possible paths.