can random forest project/interpolate based on new values of X? Sometimes I want a model to predict what would happen when presented with values of predictor variables that it has not seen before.  
For example, say, I have predictor variables (X) that go from 1 to 10 (integers only) and Y values also go up roughly from 1 to 10 (with some noise).  If I train a linear model or a random forest, they learn this relationship.   If I then ask the model what would happen if it X == 11 (a value of X that the trained model has never seen before), the linear model will guess 11.  The random forest model will, however, guess 10.  Similarly, if X == 4.51, the linear model will estimate a y value of 4.5, while the random forest will predict a value of 5 for the Y variable.  
index = np.arange(0, 300)
x = [2] *100 + [3] * 100 + [4] * 100  + [5] * 100
x = [[i] for i in x]
y = [i[0]+np.random.normal(0, .4) for i in x]
est = ensemble.RandomForestRegressor(n_estimators= 300, min_samples_split =5, oob_score=True)
est.fit(x, y)#[:-116]
x_forecast = [[i] for i in [6] *100]
#x_forecast = [[i] for i in [4.5] *100]
est.predict(x_forecast)

Is there a way to get random forest to make estimate values of Y that do not exactly equal the expected value of Y for the closest X in the training dataset? 
 A: RF regression does not extrapolate target values. RF does not interpolate within two adjacent feature values, but only assigns a break point as the mid point in between.
Strictly speaking if you refer to euclidean distance, RF does not necessarily obtain its prediction from the closest samples in feature space.
Your anticipation of a $\hat{y}$=11 for given sample $x_i$={11,11}, can only be theoretical as the training data does not at all support such an observation. Maybe you would observe $\hat{y}$=-500. RF is data driven and so to say very empirical. The model structure extrapolated from the data set is very 'flat'. Instead you could formulate your hypothesis of the system as an equation and predict extrapolated samples.
I have visualized a simple RF model to show the 'flatness' of inter- and extrapolation in a RF model. In this example, there is only 5 levels for each feature.
library(forestFloor)
library(randomForest)
X = data.frame(replicate(2,sample(1:5,2000,rep=T)))
y = -2 * (X[,1]-3)^2+2 * X[,2] + rnorm(2000)
rfo = randomForest(X,y)
par(mfrow=c(1,2), mar=c(2,2,1,1))
#One variable at the time 
for(i in 1:2) vec.plot(rfo,X,i,zoom=2,col=fcol(X))
#both variables plotted 3d
vec.plot(rfo,X,1:2,zoom=2,col=fcol(X))



