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binary heaps are especially useful data structures used in a variety of contexts but why would we want to use one how do they work and what problems do they solve to answer these questions let's start with a problem let's say you have some tasks to complete and each task has a priority some number representing the tasks importance one for your most important tasks two for the next tier of importance and so forth you want to organize tasks such that you can easily access your most important tasks first this way of organizing data is called a priority queue a collection of values where at any time we always want access to the most important item how would you store this data in a data structure you could use an array sorted by priority that makes it pretty quick to access high-priority items but with a fixed size array you'll run into trouble if you ever need to add a new element into your priority queue there's no space for another item another option is a linked list with each node of data pointing to the node that follows it this strategy makes it possible to add new data by rearranging those pointers but it's still not so efficient to add new data we might have to look through the entire linked list just to find the right place to add our new information this is where binary heaps come in binary heaps also consist of nodes of data but they're structured as a binary tree that means instead of each node of pointing to a single other node nodes can have up to two so-called child nodes a left child and a right child a node is allowed to have no children just a left child or both a left and right child each node in a binary heap also obeys an invariant in other words a property that will always be true in a binary heap and in particular a type of binary heap called a min the value of any node must be less than or equal to the values of its child nodes why is that important well this means the node with the smallest value must be at the top or root of this tree so with this structure it's easy to find the minimum value node just by looking at the root of the tree and ignoring everything else within this binary heap there are two common operations will likely need to perform the first is insertion adding a new node to the heap as we might do if we have some new task to complete for example the second is deletion removing whatever is at the root of the heap as we might do if we've completed the highest priority task so let's start with insertion when we want to add a new value to a binary heap there are two factors we'll need to bear in mind first what...