Prove that the graph is a valid tree in Python
Before going in depth of your algorithm, a few stylistics notes:


I assume that the class Solution(object) thing is required by leetcode, so it might not apply. But as a general note, your validTree method does not need to store/interact with any state. Thus there is no need for it to be a method and would benefit to be defined as a function. Also the naming should be snake_case as per PEP8: def valid_tree(n, edges):. is_valid_tree might also be a better name.
In Python 2, the keys method of dictionaries build and return a new list. It is generally recommended that your use iterkeys instead which return a proxy object to the actual keys of the dictionary. However, such object does not support the len protocol. But the dictionary support it without having to build anything new: len(visited_nodes).


Tidy up your special case

Your algorithm performs two checks:


Every node is reached through an edge;
Every node can reach each other through at most one path.


The only special case to that is, for only one node, to have no edges. The other cases will be handled by the for loops (which will do nothing if edges is empty) and the last check. Thus I would write:

def is_valid_tree(n, edges):
    if n == 1:
        # 1 node must not have edges
        return not edges
    ...


You can also remove all special cases by using default values in visited_nodes. More on that later.

Don't use stuff that you don't use

Sounds weird, doesn't it? But the same goes for your use of defaultdict: you use it without using the extra functionalities over dict. In fact, using visited_nodes = {} will not change anything in your program. So use that instead, it will remove a bit of overhead.

defaultdicts usage is to make use of default values instead of having to explicitly assign them, such as in:

d = defaultdict(list)
d['test'].append('something')
d['test'].append('something else')
d['other test'].append('a third thing')
print(d) # defaultdict(&lt;class 'list'&gt;, {'test': ['something', 'something else'], 'other test': ['a third thing']})


But instead, here you explicitly assign empty lists before assigning to them. This is because you explicitly want to define the key of the dictionary; and, in fact, you do not care about the value. Since you do not use values, why using a dictionary to start with? Use a set they are like dictionaries, but only for keys:

def is_valid_tree(n, edges):
    if n == 1:
        return not edges

    visited_nodes = set()
    count = True
    unvisited = []

    for key, value in edges:
        if key in visited_nodes and value in visited_nodes:
            return False

        if count:
            visited_nodes.add(key)
            visited_nodes.add(value)
            count = False
            continue
        if key in visited_nodes:
            visited_nodes.add(value)
        elif value in visited_nodes:
            visited_nodes.add(key)
        else:
            unvisited.append([key, value])

    for key, value in unvisited:
        if key in visited_nodes and value in visited_nodes:
            return False

        if key in visited_nodes:
            visited_nodes.add(value)
        elif value in visited_nodes:
            visited_nodes.add(key)
        else:
            return False

    return n == len(visited_nodes)


Remove "duplicated" checks

First of, the count variable is useless: it only store whether the dictionary (or the set) is empty. You can get the exact same information using not visited_nodes which is True when visited_node is empty and False otherwise.

Second, you want to add both key and value as key (so maybe they need to be renamed) of visited_nodes if either of them is already one. You can thus simplify the writing using a boolean or and retain the same functionality:

if key in visited_nodes or value in visited_nodes:
    visited_nodes.add(key)
    visited_nodes.add(value)


Using the set approach also help you write less verbose code: use the intersection &amp; operator of sets:

for edge in edges:
    edge = set(edge)
    commons = visited_nodes &amp; edge

    # If a path exist between the nodes in edge
    if commons == edge:
        # A cycle exist and thus it is not a tree
        return False

    # If a node has been visited or if it is the first edge
    if commons or not visited_nodes:
        # Add the path
        visited_nodes.update(edge)
    else:
        # Process later
        unvisited.append(edge)


Third, as stated earlier, you can get rid of all your special cases by initializing visited_nodes = {0}. This will allow you to remove the first check (if n == 1) since it is now handled by return n == len(visited_nodes). And it will also handle adding branches from the node 0 instead of any two first nodes.

Last, the body of both for loops is pretty similar. You may want to factorize it out. Unfortunately, I'm not able to come up with something elegant. It is possible to retry edges by appending them again at the end of edges instead of using unvisited, like so:

def is_valid_tree(n, edges):
    visited_nodes = {0}

    for edge in edges:
        branch = set(edge)
        commons = visited_nodes &amp; branch

        # If a path exist between the nodes in edge
        if commons == branch:
            # A cycle exist and thus it is not a tree
            return False

        # If a node has been visited already
        if commons:
            # Add the branch
            visited_nodes.update(branch)
        else:
            # Process later
            edges.append(edge)

    return n == len(visited_nodes)


Unfortunately this can lead to infinite loops in cases where your code would have reached that return False in the second for loop. However, we can come up with some checks to prevent that. And it's still better than your solution because…

It appears that you have a bug

Calling Solution().validTree(5, [[0, 1], [3, 4], [2, 3], [1, 2]]) returns False. This is because you delay checking for [[3, 4], [2, 3]] but, when running the second checks, instead of delaying [3, 4] which would be integrated in a third pass, you return False. Same would be true for inputs such as Solution().validTree(n, [[0, 1], [n-2, n-1], [n-3, n-2], ..., [1, 2]]) which would need n-3 delayed passes to come to the answer True.

All in all, this doesn't sound like \$O(n)\$ worst case, and you need to fix that bug. Either you try something different, or you patch the previous simplification. I’d go with this one and use a Counter to track the number of time we delayed the treatment for a given edge. Special care has to be taken to provide hashable objects to the counter:

from collections import Counter

def is_valid_tree(n, edges):
    visited_nodes = {0}
    delays = Counter()

    for edge in edges:
        branch = set(edge)
        commons = visited_nodes &amp; branch

        # If a path exist between the nodes in edge
        if commons == branch:
            # A cycle exist and thus it is not a tree
            return False

        # If a node has been visited already
        if commons:
            # Add the branch
            visited_nodes.update(branch)
        else:
            edge = tuple(edge)
            delays[edge] += 1
            if delays[edge] &lt; n:
                # Process later
                edges.append(edge)
            else:
                # Too much delay, can't be processed
                return False

    return n == len(visited_nodes)