Welcome to an exciting exploration of Advanced Recursion Techniques! Recursion, at its core, is a problem-solving approach where a function calls itself to solve smaller instances of the same problem.

In this lesson, we'll extend your understanding of recursion, diving into methods that allow you to tackle complex problems, particularly those involving deep tree structures and backtracking — common in technical interviews and essential in algorithmic problem-solving.

To introduce the power of these techniques, let’s look at a recursive function that generates all permutations of an array. Here, we use backtracking, a method that explores all possible solutions by recursively building on each choice and resetting states to find alternative paths.

In our example, we generate permutations by swapping elements recursively: for each position in the array, we swap the element with every other element in sequence, moving deeper in the recursion at each step. When we reach the end of the array, we save the current state as a result.

Ruby
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1def permute(nums)
2  result = []
3  
4  # Define the backtrack method, with `first` as the starting index
5  backtrack = lambda do |first = 0|
6    if first == nums.length
7      result << nums.dup  # Save a copy of the current permutation
8    end
9
10    (first...nums.length).each do |i|
11      # Swap elements at `first` and `i`
12      nums[first], nums[i] = nums[i], nums[first]
13      # Recursively call backtrack for the next index
14      backtrack.call(first + 1)
15      # Revert the swap to reset the state
16      nums[first], nums[i] = nums[i], nums[first]
17    end
18  end
19  
20  # Start the backtracking process
21  backtrack.call
22  result
23end
24
25puts permute([1, 2, 3]).inspect
26# Output: [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 2, 1], [3, 1, 2]]

With this taste of advanced recursion, it’s time to dive deeper and solve real-world problems using these techniques. As you work through the exercises, you'll sharpen your skills and gain the confidence to tackle even the most complex recursive challenges.

Remember, mastering recursion comes through practice and understanding, so let’s jump right in!