Adding piecewise polynomials in MATLAB

I need to add piecewise polynomials derived from multiple datasets. Is there an easy way to add piecewise polynomials together without interpolating? In other words, given PP1 and PP2, is there a way to generate PP3 (where PP3 remains in a piecewise polynomial form)? e.g...

    t1 = linspace(0,1,5);
    t2 = linspace(0,1,7);
    pp1 = spline(t1,sin(pi*t1));
    pp2 = spline(t2,t2.^2);
    close all
    hold on
    tnew = linspace(0,1,50);
    h(:,1) = plot(tnew,ppval(pp1,tnew));
    plot(t1,ppval(pp1,t1),'bs')
    h(:,2) = plot(tnew,ppval(pp2,tnew));
    plot(t2,ppval(pp2,t2),'rs')
    h(:,3) = plot(tnew,ppval(pp1,tnew)+ppval(pp2,tnew));
    legend(h,{'spline of sin(\pi t)','spline of t^2','sin(\pi t)+t^2'},...
                'location','northwest')
    xlabel('t')

But instead of specifying tnew explicitly, I would like a new piecewise polynomial pp3 that is effectively pp1+pp2.

This is probably the easiest way to get pp1 + pp2 Adding to the code in the question:

    pp12 = @(x) ppval(pp1,x)+ppval(pp2,x);

    breaks = unique([pp1.breaks,pp2.breaks]);
    pp3 = spline(breaks,pp12(breaks));

    plot(tnew,ppval(pp3,tnew),'k:');

Gives the dotted black line:

with pp3 being in piecewise-polynomial form with segments defined only by the breakpoints of pp1 and pp2. Running max(abs(ppval(pp3,tnew) - pp12(tnew))) yields 2.7756e-16, which is of the order of eps.

Thanks to @LuisMendo and @TroyHaskin for their suggestions.