In short, yes, you can use decision trees for this problem.
However there are many other ways to predict the result of multiclass problems.
If you want to use decision trees one way of doing it could be to assign a unique integer to each of your classes. All examples of class one will be assigned the value y=1, all the examples of class two will be assigned to value y=2 etc. After this you could train a decision classification tree.
Here is a quick implementation in Python which I got by modifying the example in this hackernoon post (https://hackernoon.com/a-brief-look-at-sklearn-tree-decisiontreeclassifier-c2ee262eab9a)
You can see that we have classes 0,1,2 and 3 in the data and the algorithm trains to be able to predict these perfectly (note that there is over training here but that is a side note)
from sklearn import tree
from sklearn.model_selection import train_test_split
import numpy as np
features = np.array([
[29, 23, 72],
[31, 25, 77],
[31, 27, 82],
[29, 29, 89],
[31, 31, 72],
[29, 33, 77],
]*10)
labels = np.array([
[0],
[1],
[2],
[3],
[2],
[0],
]*10)
X_train, X_test, y_train, y_test = train_test_split(
features,
labels,
test_size=0.3,
random_state=42,
)
clf = tree.DecisionTreeClassifier()
clf.fit(X=X_train, y=y_train)
clf.feature_importances_ # [ 1., 0., 0.]
clf.score(X=X_test, y=y_test) # 1.0
clf.predict(X_test) # array([0, 0, 0, 3, 1, 0, 3, 0, 0, 3, 2, 2, 1, 3, 2, 0, 2, 0])
