\section{Introduction}\label{sec:intro}
In their seminal paper \cite{BalancedTableaux}, Edelman and Greene introduced \emph{balanced tableaux} and showed that they are equinumerous to standard Young tableaux of the same shape. They defined the \emph{Edelman-Greene insertion} which yields a bijective proof of the reduced words of the longest permutation being equinumerous to standard Young tableaux of staircase shape, a result due originally to Stanley \cite{stanley1984number}. Fomin, Greene, Reiner and Shimozono \cite{fomin1997balanced} later generalized this enumeration result to diagrams and related the story to Schubert polynomials. 

Shifted tableaux, just as Young tableaux, are also algebraically and combinatorially meaningful (see for example \cite{sagan1987shifted,WorleyThesis}). In this paper, we define \emph{balanced shifted tableaux} (\cref{def:balanced}), as an analogue to balanced tableaux,
%using certain rank conditions on the hooks, 
from the perspective of root systems of type $B$ and $C$. The following is our main theorem, which says that balanced shifted tableaux are equinumerous to standard shifted tableaux. 
\begin{theorem}\label{thm:main}
For a shifted shape $\lambda$, the number of standard Young tableaux of shape $\lambda$ equals the number of balanced shifted tableaux of shape $\lambda$. 
\end{theorem}
We prove \cref{thm:main} by presenting an explicit bijection between the two sets of objects, $\SYT(\lambda)$ and $\BS(\lambda)$. Specifically, we have the following chain of bijections: 
\[\SYT(\lambda)\longleftrightarrow\SYT(Z(d,r))|_{\lambda}\longleftrightarrow\Red(w^{(d,r)})|_{\lambda}\longleftrightarrow\BS(Z(d,r))|_{\lambda}\longleftrightarrow\BS(\lambda),\]
where we address each step separately. We defer the definition of $\SYT(\lambda)$ and $\BS(\lambda)$ to \cref{sec:prelim} and the definition of $\SYT(Z(d,r))|_{\lambda}$, $w^{\lambda}$ and $\BS(Z(d,r))|_{\lambda}$ to \cref{sec:general}.
Here, $\SYT(\lambda)\rightarrow \SYT(Z(d,r))|_{\lambda}$ and $\BS(\lambda)\rightarrow\BS(Z(d,r))|_{\lambda}$ are the procedures to pad a tableau from shape $\lambda$ to a large trapezoid $Z(d,r)$, while the middle steps utilize type $B$ Edelman-Greene insertion defined by Kra\'skiewicz \cite{K89}.
Our strategy largely follows the framework of Edelman and Greene \cite{BalancedTableaux}, with the main difference that double staircases, which are the analogues of staircases in type $B$, are no longer sufficient for padding purposes.

% The remainder of the paper is organized as follows. In Section~\ref{sec:prelim}, we introduce key definitions and necessary background. In Section~\ref{sec:trapezoid}, we discuss the trapezoid shape $Z(d,r)$ which serves the purpose of the staircase shape in type $A$, and provide a bijection between $\BS(Z(d,r))$ and $\Red(w^{(d,r)})$, where $w^{(d,r)}$ is certain signed permutation. In Section~\ref{sec:kraskiewicz}, we provide a bijection between $\SYT(Z(d,r))$ and $\Red(w^{(d,r)})$ via Kra\'skiewicz's insertion algorithm, thus establishing the main theorem for the trapezoid case. Finally in Section~\ref{sec:general}, we finish the proof of Theorem~\ref{thm:main} for general shifted shapes by restricting the above bijections to $\lambda\subset Z(d,r)$.