One hot encoding and its combination with DecisionTreeClassifier
The two options you are describing do two very different things. 
If you choose to binarize (one-hot encode) the values of the variable, there is no order to them. The decision tree at each split, considers a binary split on each of the new binary variables and chooses the most informative one. So yes, each binary feature is now treated as an independent feature.
The second choice puts the values in an order. Implicitly you are saying that a < b < c if transforming a=1, b=2, c=3. If you use this variable in a decision tree, the algorithm considers the splits 1 vs 2,3 and 1,2 vs 3 (but not 3 vs 1,2). 
So the meaning of the variables is very different and I don't think you can expect equivalent results.
So my understanding is that you perform one hot encoding to convert categorical features as integers to fit them to scikit learn machine learning classifier.
So let's say we have two choices
a. Splitting all the features into one hot encoded features (if A is say a categorical features that takes values 'a', 'b' and 'c', then it becomes A_a, A_b and A_c with binary values in each of its rows with binary value '1' meaning that the observation has the feature and binary value '0' meaning it does not possess the feature!). I would then fit a DecisionTreeClassifier on this.
b. Not splitting all the features, but converting each category into an integer value WITHOUT performing one hot encoding (if A is say a categorical features that takes values 'a', 'b' and 'c', then 'a', 'b' and 'c' are renamed as 1, 2, 3 and no new columns are created, 'A' remains a single column with integer values 1, 2, 3 by using pandas.factorize or something which you an then fit a DecisionTreeClassifier.
My question is, when you fit DecisionTreeClassifier on the one hot encoded dataset, with multiple columns, will each of the new columns be treated as a separate feature?
Also, if you fit the DecisionTreeClassifier on the dataset where the categorical features are simply converted to an integer and kept in a single column; will it produce the same node splits, as the one where the DecisionTreeClassifier was fit on the dataset with the one-hot encoded features?
Like, when you visualize the tree in both cases, 
is the interpretation given below the right way to look at it?
for DecisionTreeClassifier with one-hot-encoding
if attribute == A_a, then yes
if attribute == A_b, then no
for DecisionTreeClassifier without one-hot-encoding ('a' represented by integer value 1 and 'b' by value 2)
if attribute == 1 then yes
if attribute == 2, then no